Overview
TL;DR - This is organized so you can skip down to what you need to know.
Here is what I'll cover below:
- QUICK OVERVIEW - What the 4 types of branches are and where you find them
- SHORT GLOSSARY - Basic principles and terminology related to branches
- INVESTIGATION - How to inspect your local and remote branches
- RELATED FILES - Configuration files
- CONFIGURATION - How to see and setup your branch configuration
- COLLABORATING - How to use remote branches
QUICK OVERVIEW
A local branch is a name, on our local repository, that refers to a head here.
A remote branch is a name, on a remote repository, that refers to a head there.
A simple branch is a local name that references one thing:
- it directly points to a local head, (i.e. to a specific commit; the growing tip)
A tracking-branch is a local name that references two things:
- it directly points to a local head, (i.e. to a specific commit; the growing tip), and
- it symbolically points to a second branch on a remote repository.
There are two kinds of tracking-branches:
- local - where the branch points to a local head.
 These are called local-tracking-branches.  (More on these below.)
 
- remote - where the branch points to a local copy of a remote head.
 These are called remote-tracking-branches.  (More on these below.)
 
Here are the 4 types of branches, where we see them, and how they map:
WHERE    ---BRANCH TYPE--------     --REFERENCE TARGETS-------
--------------------------------------------------------------
Remote   simple branch -----------> remote head (a commit ID)
--------------------------------------------------------------
Local    simple branch -----------> local  head (a commit ID)
Local    local  tracking-branch --> local  head (a commit ID1)
                                --> Remote-name/branch-name
Local    remote tracking-branch --> local  head (a commit ID2)
                                --> Remote-name/branch-name
--------------------------------------------------------------
SHORT GLOSSARY
The terms Remote and Branch appear to be overloaded.
And the phrase tracking branch is especially confusing, because it's not really the same thing as a tracking-branch.
'a snapshot'      - A recording of the state of one or more files 
                    and their contents at a given moment in time.
'a commit'        - A container holding one snapshot, the date and
                    time it was  recorded, who recorded it, and a
                    comment to say what it's all about.
'a repository'    - A repository of commits, organized so we can 
                    look thru them, going backwards in time.
                    Much like photos added in sequence to a photo
                    album book, to record our own history, each commit
                    contains a snapshot of the exact state of our
                    project at a given moment in time.
                    
                    It is used to be able to look backwards in time to
                    how it was at any recorded previous time.
'Remote'          - (Upper case) Short for 'a named remote repository'
                                 (of commits, of snapshots)
'remote'          - (Lower case) Located on  another     git repository
'local'           -              Located on  your local  git repository  
'a head'          - A specific young commit, with no children yet of
                    it's own (i.e. no other commits yet pointing to it),
                    but which may link backwards in time to one or more
                    of its natural parents.
                    Also called a growing tip.
                    Initially set to a <start-point>. 
'a branch'        - A symbolic name (i.e. an identifier) pointing
                    to one specific head, and possibly, depending on
                    the branch type, also pointing to a remote branch.
                    The term 'branch' can also refer to a specific
                    linked list of multiple commits (plural), starting 
                    from the growing tip (or most recent baby), and 
                    linking offspring to their parent(s) backwards in
                    time.
                   
'tracks'          - As we move forward, tracks are what we leave behind.
'tracked'         - To be followed, as in, to come afterwards, or after
                    the fact, by way of the evidence left behind, of the
                    a state of being of the thing being tracked, as it
                    moves forwards in time.
'tracking'        - The process of capturing and organizing snapshots of
                    our project so we can later look backwards in time
                    to find how it previously was.
'tracking-branch' - This term is somewhat redundant, and confusing, 
                    but does have a specific, important meaning.
                    I have deliberately added the hyphen, because this
                    term does NOT mean simply 'tracking branch'.  (Grab
                    your aspirin, and a cold pack for your head, lol.)
                    Because all branches in git are used for, and only
                    used for, tracking your project, therefore it could
                    be said that ALL branches are actually 
                    'tracking-branches', but we don't call them that.
                    Instead we call them, simply 'branches'.
                    But then what is a 'tracking-branch'?
         TL;DR      A 'tracking-branch' is a local name that points to
                    two branches at the same time.
                    
                      So when you read  'tracking-branch,  it might be 
                      helpful to instead think:  'branch-pair'.
                        (Normal branches only point to one thing, the
                        head, which is the commit at a growing tip.
                        And they do not have any symbolic pointers.)
                    1) The first branch a 'tracking-branch' points to
                    is the same as for any other branch:  a local head,
                    (i.e. a young commit in our local repository without 
                    any children.)  This is where a tracking-branch
                    keeps a full local copy of a remote branch.
                      Note that it doesn't necessiarialy hold a full
                      duplicate copy of the entire second, remote 
                      repository.  If you have cloned the remote 
                      repository then you already have most, if not all
                      of their commits in your own local repository.
                    2) The second branch a 'tracking-branch' points to
                    is a branch on a remote repository.
                      It does this with a <remote-name>/<branch-name>.
                      The 'remote-name' is used to find the URL to the 
                      remote repository.  See `git remote -v`.
                    Why point to two branches?  
                      This is to be able to operate on two heads at the
                      same time, like to copy commits from one head to
                      the other as `git fetch` and `git push` does.
                    We have two types of 'tracking-branches' (both on
                    our local repository):
                    
                      'local  tracking-branches', 
                           with a simple     branch name,  and
                           
                      'remote tracking-branches', 
                           with a path-style branch name.
                           
                      See `git branch -avv`.  For example:

- The 1st two lines output here, are local tracking-branches.  The asterisk (*) prefixing - mastertells us that- masteris currently the default branch (i.e. what is checked out into our working area).  BTW, the name- masteris short for- refs/heads/master.
 
- The 3rd line output is a simple local branch. 
- The 4th line output is NOT a branch at all, but rather a second local HEAD (in addition to our normal local HEAD) which points to the default remote tracking-branch, or one of the following branches in this example.  Use - git remote set-head <remote-name> <remote tracking-branch name>to set it.  (NOTE this is also not the same HEAD as returned by- git remote show <remote-name>which is the downloaded value of the remote repository's HEAD.)
 
- The last two lines output are remote tracking-branches. 
Note that all branches reference a commit ID (hex number). remotes/origin/HEAD is not a branch so it doesn't have this.
Also note that the first two lines and the last two lines also have a symbolic reference to a branch on a remote (in this case the remote named origin).
Here 'master' is our local working branch.  And remotes/origin/master is a local copy of the branch named master fetched (by git fetch, git clone or git pull) from the remote we call origin.
(BTW, origin is the default name of the Remote repository that we originally cloned, with a git clone command.)
                    So our 'remote tracking-branches' are not remote 
                    branches, on a remote repository, but rather are 
                    local branches, which have a local head of their
                    own, pointing to a local commit, and also at the 
                    same time symbolically pointing, to a remote 
                    branch.
                    With `git branch -avv`, notice how two branches can
                    point to origin/remote:
                    
                    * the first  being the  'local-tracking-branch' 
                      with the name        'master', and with the
                      '[origin/master]' extra clause,  and 
                      
                    * the second being the 'remote-tracking-branch'
                      with the name 'origin/master'.
                    NOTE: Though they point to the same remote branch, 
                    the local commit head is not always the same!
                    
                    Thus they are actually two different branches.  
                    
                    The 'local-tracking-branch' is our working branch, 
                    and the 'remote-tracking-branch' is a copy of the 
                    remote's branch that we cloned from or fetched to
                    update.
INVESTIGATION
REMOTES
git remote                      # List names of known Remotes  
git remote -v                   # List names of known Remotes and
                                #   show the 2 URL's pointing to them  
                                #
                                #  See '[remote "<names>"]' in
                                #    $ cat .git/config
REMOTE BRANCHES (located on the remote repository)
git remote show <remote-name>   # Download and view 
                                #   a specific Remote's info.
# for example, let's download the information for 
# two remotes named origin and upstream:

- The leading asterix (*) is a bullet to mark the start of data from a given remote.  We requested downloads from two remotes, so we have two bullets. 
- The 1st line output gives the name of the remote, prefaced with the word 'remote'. 
- The 2nd and 3rd lines report our locally configured fetch and push URLs for the remote named - origin.  View them also with- git remote -v.
 
- The 4th line reports the HEAD from the remote repository.  You can't set this HEAD.  Nor is it the same as the local HEAD, nor local read for remotes from - git branch -avv
 
- Starting on the 6th line is the list of branches that the remote repository owns. - Remote branches:
master  tracked
updates tracked 
- Then torek says this about the remaining lines: 
All git remote show does is call [the remote] up, using git ls-remote, over the Internet-phone, and compare their references to your references to guess what git fetch and git push would do, based on those results.  (If you use git pull, that just means run git fetch, then run git merge. The git remote show command tries to guess what that will do, too.)
LOCAL BRANCHES (located on the local repository)
git branch -avv  # Show ALL  'local branches', verbosely;  (3 types):
git branch -rv   # -- type 1 -------------------------------------
                 # Show ONLY 'local branches' that point to
                 # 'remote branches' (-r = remote; -v = verbose)
                 #
                 #   This lists your 'Remote tracking branches'!
                 #     From:  $ tree .git/refs/remotes/*
                 #
                 #      They allow us to move snapshots between
                 #       repositories, and to keep a copy of
                 #       Remote's branches locally.
git branch -vv   # -- types 2 and 3 ------------------------------
                 # Show ONLY 'local branches', that point to local
                 # things, but his includes two different types of
                 #  branches mixed together, for example:
* master  de430b6 [origin/master] <comment describing this branch>
  updates 3c40299 [origin/updates] <comment describing this branch>
  foo     de430b6  <comment describing this branch>
Notice that the first two branches named master and updates (above), both have an additional field after their commit number.  For example, for the branch named 'master', this field is [origin/master].
This tells us that these two branches are not ordinary local branches, but rather are Local tracking-branches.  Similar to the 'remote-tracking-branches' above, they also symbolically point to a remote branch.  Thus master in this case, not only points to a branch head in the local repository, but it also points to origin/master, in the remote repository.
These extra fields are setup by parameters in .git/config.
By contrast, foo here is a simple, normal branch, i.e. non-tracking.
RELATED FILES
 cat .git/config                       # '[branch "<names>"]' are local
                                       #    tracking branches
 ls -F .git/refs/heads/*               # 'Local' branch names & heads:
                                       #   both tracking and non-tracking
 ls .git/refs/remotes/<remote-name>/*  # 'Remote' tracking branch names & heads
CONFIGURATION
Create with git branch, git checkout -b, or
by cloning a remote repository with git clone, or
manage explicitly either by directly edting .git/config or with these:
Remotes
Create with git remote implicitly by cloning a git repository with
git clone.
- git remote add- to explicitly add a new remote name (to .git/config)
- git remote rename
- git remote remove- to delete a remote
- git remote prune- to delete any local remote-tracking branches which have already been removed on the remote
Set properties with:
- git set-url- set one url, or replace a url for the remote
 
- git set-url --add- append a url, to the list of urls for the remote
 
- git set-url --delete- remove all url's matching a pattern
 
- git set-branches- changes the set of tracked branches
 
- git set-branches --add- append, rather than fully replace the list of currently tracked branches
 
- git set-head- sets the default remote branch (i.e. the remote's HEAD)
 
- git set-head --auto- query remote to set the local HEAD for the remote branch
 
- git set-head --delete- deletes the default remote branch (i.e. the remote's HEAD)
 
Branches
git branch  [--set-upstream | --track | --no-track]  [-l] [-f]               <NewBranchName> [<start-point>]   # create branch         (start point defaults to HEAD)
git branch  (--set-upstream-to=<upstream-branch> | -u <upstream-branch>)       [<BranchName>]                  #   link to upstream branch
git branch --unset-upstream                                                    [<BranchName>]                  # unlink to upstream branch
git branch --edit-description                                                  [<BranchName>]                  # edit   branch description
git branch (-m | -- move | -M)                              [<oldBranchName>] <newBranchName>                  # rename (move) branch; -M = force
git branch (-d |           -D) [-r]                                             <BranchName>...                # delete branch
COLLABORATING
With a default configuration, when you git clone this automatically sets your remotes and tracking-branches.  Note however, that there are configuration settings that will disable or change how this works.
TIP Use the --dry-run option on git fetch and git push to see what's going to happen before you do it.
Use git fetch (possibly by calling git pull) to update your local copy of a remote's commits to bring you up to date.
If you don't include a  then the default is used.  You can see what the default is in .git/config in the fetch= property under [remote "<remote-name>"].  This might look like this:
[remote "origin"]
    fetch = +refs/heads/*:refs/remotes/origin/*
Syntax is [+]?<source>:<destination>.  What that means is to get the refs (normally commits and tags) from .git/refs/heads/* which are normal, simple branches in the remote repository, and put them into our local .git/refs/remotes/origin/* branches, which are our tracking-branches.  Cool, eh!  BTW, the '+' says to update even if this won't be a fast forward.
Use git push <remote> <branch> to send local commits to a remote repository that you have permission to write to.
I hope I've got this all correct.