Quads are not always necessary, but are usually preferred when deforming a mesh for animation &/OR Rigging. If you are creating a static model (eg. a sculpture of a statue, etc...) Then it can be acceptable to use Triangulated faces, as there is no reason to worry about the integrity of the deformations.
Notice the deformation smoothness when we bend a cylinder.
As for my version of face Definitions:
NGon technically a face with vertices > 3, but because we have names for 3 vert, and 4 vert faces, this is commonly reserved to describe a face >= 5.
Quad = 4 vertices.
Triangulated is just the primitive minimum for a face to exist, as the vertices that make it up = 3.
Two vertices make an edge with no area.
One is just a single point (Vertex).