Throwing away your local changes
If you want to discard your work, fetch and reset.  For example, if you have a remote named origin and a branch named master:
$ git fetch origin
$ git reset --hard origin/master # Destroys your work
Keeping your local changes
If you don't want to throw away your work, you will have to do a git rebase --onto. Suppose the old origin looks like this:
A ---> B ---> C
              ^
              origin/master
And you have this:
A ---> B ---> C ---> X ---> Y ---> Z
              ^                    ^
              |                    master
              origin/master
Now, the upstream changes change things:
A ---> B ---> C ---> X ---> Y ---> Z
 \                                 ^
  ---> B'---> C'                   master
              ^          
              origin/master
You would have to run git rebase --onto origin/master <C> master, where <C> is the SHA-1 of the old origin/master branch before upstream changes.  This gives you this:
A ---> B ---> C ---> X ---> Y ---> Z
 \
  ---> B'---> C'---> X'---> Y'---> Z'
              ^                    ^
              |                    master
              origin/master
Notice how B, C, X, Y, and Z are now "unreachable". They will eventually be removed from your repository by Git.  In the meantime (90 days), Git will keep a copy in the reflog in case it turns out you made a mistake.
Fixing mistakes
If you git reset or git rebase wrong and accidentally lose some local changes, you can find the changes in the reflog.
In the comments, a user is suggesting git reflog expire with --expire=now but DO NOT RUN THIS COMMAND because this will DESTROY your safety net.  The whole purpose of having a reflog is so that Git will sometimes save your neck when you run the wrong command.
Basically, what this command will do is immediately destroy the B, C, X, Y, and Z commits in the examples above so you can't get them back.  There's no real benefit to running this command, except it might save a little bit of disk space, but Git will already purge the data after 90 days so this benefit is short-lived.