AA and AF are settings designed to enhance the visual quality of edges and textures, respectively.
AA (Anti-Aliasing) seeks to eliminate jagged edges on 3d models and other images by rendering an image at multiple times its original size, and recompressing the image down to the size intended for display on the monitor. For instance, if you have 16x AA enabled, your GPU is rendering the image at 16 times it's size on the monitor and recompressing it down to size in real time to eliminate the jagged edges.
AF (Anisotropic filtering) seeks to eliminate texture aliasing (blurry textures in the distance) when viewing textures from extreme angles. This is done by taking samples of the texture and using that data to improve the quality of the texture when applied to an oblique surface. 16x AF requires approximately 128 samples from a texture to apply the filtering.