Huh, you're mixing two things together...
Texture filtering - the main purpose of this is to reduce aliasing of applied texture on image. For the purpose of this we use texture magnification filtering (aka when 1 pixel in texture covers more pixels on screen), and texture minification filtering (aka when several pixels in image covers 1 pixel on screen).
The minification filtering is often done using MIP maps (multum in parvo) - I guess you know the principle, just put it into wiki and read how it works.
Texture compression is something else, this relates to texture storage. You can store textures are RGB8 (8-bits per channel), RGB32F (32-bit float per channel), R5G6B5 (5 bits red and blue, 6 bits red), S3TC DXT1 aka BC1 (this one is compressed using S3 texture compression algorithm - this is compressed texture format).
OpenGL supports hardware decompressing of several formats (F.e. named S3TC), and it can also compress these textures. The quality of compressed textures by hardware is low, so it is recommended to compress using other tools (I use my own), but F.e. The Compressonator from AMD works perfectly.