I sort of recommend the "search" function, as this has been discussed many times before.
Both AA and AF use filtering techniques to make the rendered image from the video card look better. AA (Antialiasing) helps smooth out "jaggies" -- aliasing artifacts created when diagonal lines are drawn on a computer screen. AF (Anisotropic Filtering) removes certain filtering artifacts that are created when you view textured surfaces at an angle.
They both give you a performance hit. On ATI cards, low levels of AF are essentially "free" -- in most games, the video card has enough spare bandwidth that turning it on doesn't really slow anything down. However, even 2xAA has a noticeable impact on performance (and visual quality).
I run my 9800Pro (430/375) at 4xAA/16xAF, usually. 6xAA seems to make little visual difference in most games, but hurts framerates. To each his own, though.