An old chestnut goes like this:
 ___ ___
|   |   |
|   |   |
|___|___|
|   |   |
|   |   |
|___|___|
A 2x2 grid of squares are formed by 12 matchsticks as above. Remove two matches to leave just two squares and no extraneous matches that are not part of any square.

Yes, it is really possible to do this! If you think it's impossible, reexamine your assumptions.

There are a number of other closely related puzzles, such as:

 ___ ___ ___
|   |   |   |
|   |   |   |
|___|___|___|
|   |   |   |
|   |   |   |
|___|___|___|
|   |   |   |
|   |   |   |
|___|___|___|
Remove just 4 of the 24 matches to leave only 6 squares.

Answers

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