I would like to ask about examples where experimentation by computers have led to major mathematical advances.
A new look
Now as the question is five years old and there are certainly more examples of mathematical advances via computer experimentation of various kinds, I propose to consider contributing new answers to the question.
I am aware about a few such cases and I think it will be useful to gather such examples together. I am partially motivated by the recent polymath5 which, at this stage, have become an interesting experimental mathematics project. So I am especially interested in examples of successful "mathematical data mining"; and cases which are close in spirit to the experimental nature of polymath5. My experience is that it can be, at times, very difficult to draw useful insights from computer data.
Meta Question: classification of experimental mathematics
I asked: Is there a useful way to classify the various types of experimental mathematics?
Let me try to propose, based on the answers so far, a tentative answer describing five categories of experimental mathematics:
1) Mathematical conjectures that were arrived at by examining experimental data
2) General purpose programs which interactively or automatically lead to posing mathematical conjectures.
3) Large mathematical databases
4) Computer-assisted proofs of mathematical theorems
5) Various computer programs which allow proving automatically theorems or generating automatically proofs in a specialized field.
There were many excellent answers so let's give the bounty to Gauss...