Anti Aliasing, like Martin said merges pixels that are close together which removes the jagged edges in objects that are curved or diagonal. Ij you were to take a close look at a curve for instance, you would see several things like this _l making the circle. Anti aliasing would blend those 2 pixels to make it more like this /. The higher the setting, the more samples you take and blend together to improve the effect. This is very demanding on your CPU since it has to do several calculations.
Anisotropic filtering will increase the LOD (level of details). This is mostly visible in very close objects. A good example is with the Star-Trek Elite Force 2 games. With AF set to 4X you could read the ship's computer panels when standing close to them while they were blurry with no AF. It also requires a lot of resources so unless you have a good CPU and video cards, these 2 have to be set fairly low in order to eliminate choppy fps and gameplay.
Since lower resolution will show more jagged edges than higher ones (i.e. 800 X 600) aliasing becomes less apparent at higher resolution therefore the level of AA used can be decreased.
If you check ANANDTECH and TOMSHARDWARE site, they give good visual examples of AA and AF for both ATI and NVIDIA
My gaming machine:
P4 2.8 GHz
2 X 512 MHz PC3200 in dual DDR mode
Intel D865PERL MOBO
integrated 6.1 MAX audio
Logitech duo MX mouse and KB
ATI RADEON 9800 PRO