Warning: Wikipedia contains spoilers.
Deep Thought was created to come up with the ultimate answer to Life, the Universe, and Everything. When, after seven and a half million years of calculation, the answer finally turns out to be 42, Deep Thought's creators sheepishly realize that they don't know the question. Deep Thought itself does not know the ultimate question to Life, the Universe and Everything, but offers to design an even more powerful computer (the Earth, note the Earth in fiction) to calculate it. After ten million years of calculation, the Earth is destroyed by Vogons five minutes before the computation is complete.
In the original radio series, Deep Thought was voiced by Geoffrey McGivern who also played Ford Prefect. In the subsequent LP re-recording and television series, Deep Thought was voiced by Valentine Dyall.
The name Deep Thought is a parody on the movie Deep Throat.
Deep Thought is also a real computer, first in a line of chess computers that included Deep Blue, the computer that defeated Garry Kasparov in a six-game chess match. It was named Deep Thought in reference to Adams' fictional computer. The naming of chess computers has continued in this vein with Deep Fritz, Deep Junior, etc. ("Deep" also comes from the way the various algorithms look for all possible moves and potential outcomes.)