A pyramid is a geometric solid, having a polygon as its base (or bottom), with triangles for its faces (or sides) and a vertex that is perpendicular to the base.
The polygon base can have any number of sides, 3 or greater.

A pyramid gets its name from its polygon base and not from its faces.
So, the diagram at the top of this page depicts a square pyramid. The diagrams below show a triangular pyramid and a pentagonal pyramid.
 

Like the diagram at the top of the page, the Great Pyramid of Egypt has a square base and 4 triangular faces and that's what most people think of when they hear the word 'pyramid'.
However, mathematically speaking, the base could have any number of sides, 3 or greater.