The way I always thought of it is this:
Imagine you have two movable objects linked by a spring. When one object moves, it stretches the spring, which makes the other object move. Now, you can draw up a system of differential equations which describe all this, and numerically integrate to figure out what the system does. (Actually, a system this trivial might even admit a closed-form solution.)
Now imagine taking the spring and replacing it with a steel bar.
Scientifically, a steel bar is just like a spring - it can be stretched and squashed. But practically, the actual deformation of the bar is going to be minute. (Unless you're dealing with really huge forces.)
Basically, what you're saying is that a steel bar is just a spring with an insanely large spring constant. In other words, the steel bar is a "stiff spring".
Now think about how you're going to solve that numerically. With a "normal" spring, when you move one end, the other end oscillates around a bit and eventually settles down at the natural length of the spring. When you move a steel bar, what happens in the real world is that the other end instantly reaches equilibrium. What happens in the simulation is that the huge spring constant causes the system to oscillate like crazy, and you have to take tiny, tiny steps to make it converge to equilibrium properly.
In short, stiff systems are a pain to solve numerically. In fact, while reading an nVidia (?) tutorial on numerical integration, they gave the following advise for dealing with stiff systems: "Try to make it un-stiff. Failing that, use implicit methods. Good luck with that..."
This isn't a "definition", but should give you some idea of the intuition behind the term. (Unless, of course, I'm completely wrong... In which case I'm sure somebody will say so!)