TITLE: What's the intuition behind "stiff systems"?
QUESTION [16 upvotes]: One of the mysteries that my numerics class left me with are "stiff systems". Our prof didn't really manage to explain what they are. Now that the class (and the test) are over, I'm still curious. 
The most concise definition about "stiff systems" I've found so far is: "There is no universally accepted definition of stiffness". So, if there is none, I don't expect one. 
I'd be thankful about information on both of the following points:

From what kind of (real-world) problems do stiff systems arise? What's so "stiff" about them?
What's the good thing - or the bad thing - that I'd worry about when I have to numerically solve such a real-world problem?

REPLY [2 votes]: Imagine that the solution of the initial value problem that you are integrating, changes very slowly, but on the other side, you have nearby solutions that change very fast.
Your method of integration is approximate and is never exactly 'on top' of your solution, so small errors tend to magnify themselves. 
In front of this situation, algorithms with variable step size will try to make the step size very small to avoid this. 
This, of course, will make the process of integration very slow in the best case. In the worse case, small steps of integration will increase dramatically the rounding errors of the method.
Note that a solution of a differential equation may be stiff for some values while not stiff for other values. 
A general indicator of stiffness is a big difference in the eigenvalues of the linearized problem. Though this does not always applies it gives you an idea.
This means that your problem is scaled in different ways, so it requires small step size and large step size at the same time.