First, some US specific legal bits (be sure to check your local legislation): It's not yet illegal to have and run an emulator. The trouble is in getting the ROMs for games that you want to run on the emulator. While it's legal for game owners to create an archival backup copy of a game, it's illegal to download or distribute ROMs or image files from the net, even if you legally own a copy of the game. You have to own the game and make the copy yourself.
Another consideration is that some emulators require the console BIOS to run. Again, it is illegal to distribute this, and though it's possible to dump the console BIOS onto a memory card, downloading or distributing the BIOS file is illegal and may void warranty.
Again: It's only legal for you personally to make ROM or image copies of a game you own. It's illegal to download, share or host ROM or image files of the games themselves.
(EDITOR'S NOTE: Main source for this writeup can be found here.)