>> that somebody won't be able to
  >> disagree with by counter example

  > I really don't understand this.
  > You give a counterexample to a
  > theorem; you don't give one to
  > a definition.

In a sense you do. Often when we are giving a definition we are intending to capture a particular idea. Sometimes the definition we give captures too little, or too much. If you find that out early enough then you can change your definition to better match what you intend.

The classic example is "connected" from topology. Speaking very loosely, a set is "disconnected" is there is a "disconnection", which is a separation of the set into two pieces which are contained in disjoint open sets. Basically, the set is disconnected if it's made up of two (or more) pieces that can be divided by a "surface". A set is "connected" if there is no disconnection.

The problem is that there are sets we want to think of as not connected, and yet which satisfy the definition of connected as given above. Here's an example:

  X = { (x,sin(1/x)) : x in R, x>0 } u { (0,y) : -1 < y < 1 }

So the net result is that we have the two definitions: "connected" and "pathwise connected."

If that example had been thought of earlier, it's possible that the definition of "connected" might have been fixed earlier. So in a sense, X is a counter-example to the definition of connected.

Further reading:

http://planning.cs.uiuc.edu/node140.html

https://en.wikipedia.org/wiki/Connected_space

https://en.wikipedia.org/wiki/Connected_space#Path_connected...