What does 'Monkey Patching' exactly Mean in Ruby?

According to Wikipedia, a monkey patch is:

a way to extend or modify the runtime code of dynamic languages [...] without altering the original source code.

The following statement from the same entry confused me:

In Ruby, the term monkey patch was misunderstood to mean any dynamic modification to a class and is often used as a synonym for dynamically modifying any class at runtime.

I would like to know the exact meaning of monkey patching in Ruby. Is it doing something like the following, or is it something else?

class String
  def foo
    "foo"
  end
end

The best explanation I heard for Monkey patching/Duck-punching is by Patrick Ewing in RailsConf 2007

...if it walks like a duck and talks like a duck, it’s a duck, right? So if this duck is not giving you the noise that you want, you’ve got to just punch that duck until it returns what you expect.

The short answer is that there is no "exact" meaning, because it's a novel term, and different folks use it differently. That much at least can be discerned from the Wikipedia article. There are some who insist that it only applies to "runtime" code (built-in classes, I suppose) while some would use it to refer to the run-time modification of any class.

Personally, I prefer the more inclusive definition. After all, if we were to use the term for modification of built-in classes only, how would we refer to the run-time modification of all the other classes? The important thing to me is that there's a difference between the source code and the actual running class.

In Ruby, the term monkey patch was misunderstood to mean any dynamic modification to a class and is often used as a synonym for dynamically modifying any class at runtime.

The above statement asserts that the Ruby usage is incorrect - but terms evolve, and that's not always a bad thing.

Monkey patching is when you replace methods of a class at runtime (not adding new methods as others have described).

In addition to being a very un-obvious and difficult to debug way to change code, it doesn't scale; as more and more modules start monkey patching methods, the likelihood of the changes stomping each other grow.