extend - adds the specified module's methods and constants to the target's metaclass (i.e. the singleton class) 
  e.g. 
- if you call Klazz.extend(Mod), now Klazz has Mod's methods (as class methods)
- if you call obj.extend(Mod), now obj has Mod's methods (as instance methods), but no other instance of ofobj.classhas those methods.
- extendis a public method
include - By default, it mixes in the specified module's methods as instance methods in the target module/class. 
  e.g.
- if you call class Klazz; include Mod; end;, now all instances of Klazz have access to Mod's methods (as instance methods)
- includeis a private method, because it's intended to be called from within the container class/module.
However, modules very often override include's behavior by monkey-patching the included method. This is very prominent in legacy Rails code. more details from Yehuda Katz. 
Further details about include, with its default behavior, assuming you've run the following code
class Klazz
  include Mod
end
- If Mod is already included in Klazz, or one of its ancestors, the include statement has no effect
- It also includes Mod's constants in Klazz, as long as they don't clash
- It gives Klazz access to Mod's module variables, e.g. @@fooor@@bar
- raises ArgumentError if there are cyclic includes
- Attaches the module as the caller's immediate ancestor (i.e. It adds Mod to Klazz.ancestors, but Mod is not added to the chain of Klazz.superclass.superclass.superclass. So, calling superin Klazz#foo will check for Mod#foo before checking to Klazz's real superclass's foo method. See the RubySpec for details.).
Of course, the ruby core documentation is always the best place to go for these things. The RubySpec project is also a fantastic resource, because they document the functionality precisely.
ref: John