A quad is a four-sided Polygon.
Some game engines use quads instead of triangular polygons, but most engines just use tris.
A greater benefit for game artists is that quads work well with edge-loop selection & transform methods that speed up modeling, make it easier to judge the "flow" of a model, and make it easier to weight a skinned model to its bones.
Artists usually preserve quads in their models as long as possible. However Normal mapping usually requires a triangulated model, which keeps the original model's tangent basis consistent, so it will render properly.
The artist should try to keep the vertices of each quad Coplanar, or rendering weirdness can happen, because the quad is divided into two triangles at render time. The internal edge between the two tris is determined arbitrarily, so it could look like a ridge or a valley.
See also Polygon Count.