a quad is 2x2 pixels
The pixel shader always processes 2 X 2 groups of pixels (called "quads") but there is no other part of a GPU that necessarily does. Quads exist for the purpose of mip mapped texture filtering; if you compute texture coordinates for each pixel in a 2 X 2 quad, you can do a little bit of math on the four coordinates to determine which mip maps should be sampled.
This is one of many reasons why a GPU is poor at rendering tiny triangles. A one-pixel triangle is shaded as a 2 X 2 quad, which wastes 75% of pixel shader resources.