Since many texels will probably map to each pixel, it would be extremely expensive to perform the entire filter computation at each pixel, each frame. MIP-mapping (Multum in Parvo, or many in a small place) pre-computes part of the filter operation by storing down-sampled replicas of each texture, going from the original m x n texture map all the way to a 1 x 1 texture map. At run time, if a given screen pixel maps to many texels, we simply choose the appropriate MIP-map level in which a single texel represents all the desired texels, and perform our texture sampling on that level.
Since the pixel will project to an arbitrary place on the MIP-map image, rather than exactly onto a texel, we can use bilinear or other filtering to interpolate nearby texels. This is Bilinear MIP-mapped Texture Filtering talked about in video game hardware. Better than this, we can bilinearly filter on the two most appropriate MIP-map levels and then blend the results. This is Trilinear MIP-mapped Texture Filtering. It requires the use of eight texels, and requires seven linear interpolations, but provides an important improvement over bilinear filtering.