Bilinear and trilinear filtering basically blends textures together and reduces pixelization along the edges of textures and across stretched out textures. I know this is not a complete accurate description, but if you want to know 'basically' what they are, that description should help.
Bilinear filtering is common in even many old cards - the smoothing of the textures is also what causes that blurry effect for small textures that have been stretched far beyond their original size (think N64). However, this blurriness is preferable to pixelized graphics (think old psx)
Trilinear filtering is used with mip mapping I believe. Mip mapping creates artifacts known as mip-banding, and trilinear filtering smooths the transition between textures and removes these artifacts. Trilinear filtering samples twice as many pixels as bilinear filtering, but may require a second pass to render depending on your card.
Trilinear filtering looks better, bilinear filtering is faster.