Point filtering was used as a texture filter to determine a pixels color value from a texture when up close when the texture size itself wasn't detailed enough to fill the entire screen area covered by the texture. This filter simply used the adjacent texel which resulted in sub optimal pixelated images.
For example if you have a texture that is 128x128 pixels in size and you are running 1024x768 resolution that texture could be viewed so it would take up possibly 512x512 actual screen pixels. This would result in a lot of empty on screen pixels, or empty space, if nothing were done. Point filtering reuses a single texel (a pixel from the original texture in this case) and copies it to fill in the added space. This takes care of filling the "empty space", but still leaves heavy pixelization of textures
Bilinear filtering is now a base feature on all cards. Bilinear is accomplished by reading four adjacent texels versus the single texel read of point sampling. This is currently standard on all 3D accelerators and should be considered the bare minimum for acceptable image quality.
Using the same scenario as above, bilinear blends together the four texels around the empty space to create a blended, much smoother image to fill out what would be "empty space".
Trilinear filtering is the next step beyond bilinear filtering. Trilinear uses eight texel samples and also blends mip map levels together to smooth out the transition between mip levels. It handles this by taking four samples from each mip level resulting in smooth blending instead of the clear lines seperating mip levels present in bilinear filtering. Almost all new hardware supports trilinear filtering although some of them suffer from large performance hits.
Trilinear would result in eight texel samples being used from two different mip map levels to give the smoothest image currently available using these techniques.
