SSAA will double or quadruple the resolution and then scale it down to your monitor, usually has the best image quality but relies on the card being able to render massive resolutions.
MSAA is the most standard technique and can have a large memory footprint and depending on how much detail it needs to work can take up a considerable chunk of GPU. Saying that Keplar has surprised me at how many games I can stick 16x MSAA on and still keep games above 60FPS.
FXAA uses shader power to blur the edges and in some games does blur the textures quite a bit, it's probably the least intensive method.
TXAA is Nvidia's technique that is apparently great with fast motion, but again some people complain about blurring. Apparently it's not best judged from screenshots
SMAA is the latest and isn't very taxing on the card at all and is much better at working with likes at all angles than most.
Check out the 2nd half of this video:
http://vimeo.com/31247769