A square pyramid is a pyramid with a square base. It has 4 outer faces, all of which are triangles, that meet at the same point perpendicular to the centre of the base.
All pyramids have at least 4 faces (including the base), where the outer faces are all triangular while the base can be trilateral, quadrilateral, or any other polygon. The quadrilateral form (the square pyramid) is the most common form of pyramid, and is the form associated with the Egyptian pyramids.
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If you mean the base has 4 sides then the pyramid is a square based pyramid and has 5 corners. If you mean the pyramid has 4 faces then the pyramid is a tetrahedron with a triangular base and has 4 corners.
A square pyramid.
A square based pyramid.
A square pyramid has 1 more vertex than a triangular pyramid. A square pyramid has 5 vertices, and a triangular pyramid has 4 vertices.
Square pyramid : 5 vertices. Triangular pyramid : 4 vertices.