Is it illegal to take someone's photo without their permission?

If live in the United State and it is for use on the internet, brochures, magazines etc. or you are in some way profiting from it, then yes.

Laws on this matter vary greatly depending on what part of the world you live in. Be more specific on where, how, why, etc. to receive a more specific answer.

Be aware that usually if the person cannot be identified from the picture (i.e. photo taken from behind, photo of a child picking a flower (face is not facing the camera)), there is generally not a problem.

In most other parts of the world, if you take a picture while in Public, it's fair game.