Bilinear filtering is a texture filtering method used to smooth textures when displayed larger or smaller than they actually are.

Invented by: unknown
1st use: unknown
Mainstream use: 1996 - with 1st gen of 3D accelerator cards
Used for: increasing image quality
Replaced by: Trilinear filtering

Bilinear filtering basicaly smooth visible pixels in textures by taking 4 pixels and combining their colors by weighted average according to distance. This is basic technology on which was based succes of 1st 3D accelerators. Games with filtered textures are looking much better, than unaccelerated versions. Most early games supported both modes (software and 3D accelerated), so anyone could compare image quality with or without Bilinear filtering. 

Few early cards were not supporting Bilinear filtering. That sometimes increased their speed, but greatly reduced image quality compared to competitors. Namely:
Matrox Mystique
Matrox Mystique 220
Matrox Millennium II
Nvidia NV1
Videologic/NEC Midas 3
Videologic/NEC PCX1

 
Difference between bilinear filtered (right) and nonfiltered game (left), can be seen on these comparsion screenshots made in game Forsaken.

Another visible difference between filtered and nonfiltered game can be seen on planet surface (from X benchmark).

More info at:
Wikipedia
UnrealWiki
GameDev
Dev.mag

You can also view video below showing various versions of game Battle Arena Toshinden. Only 3Dfx version have Bilinear filtering enabled - most visible difference is on arena floor.