In fact if you write any C extensions for your Ruby projects there is really only one way to define a Module method.
rb_define_singleton_method
I know this self business just opens up all kinds of other questions so you could do better by searching each part.
Objects first.
foo = Object.new
Can I make a method for foo?
Sure
def foo.hello
'hello'
end
What do I do with it?
foo.hello
==>"hello"
Just another object.
foo.methods
You get all the Object methods plus your new one.
def foo.self
self
end
foo.self
Just the foo Object.
Try to see what happens if you make foo from other Objects like Class and Module. The examples from all the answers are nice to play with but you have to work with different ideas or concepts to really understand what is going on with the way the code is written. So now you have lots of terms to go look at.
Singleton, Class, Module, self, Object, and Eigenclass was brought up but Ruby doesn't name Object Models that way. It's more like Metaclass. Richard or __why shows you the idea here. http://viewsourcecode.org/why/hacking/seeingMetaclassesClearly.html And if the blows you away then try looking up Ruby Object Model in search. Two videos that I know of on YouTube are Dave Thomas and Peter Cooper. They try to explain that concept too. It took Dave a long time to get it so don't worry. I'm still working on it too. Why else would I be here? Thanks for your question. Also take a look at the standard library. It has a Singleton Module just as an FYI.
This is pretty good. https://www.youtube.com/watch?v=i4uiyWA8eFk