Square base and four sides that come to a point. Simple enough.
The image (above, right) is in Mexico. Although it does not come to a full point, it is considered a pyramid. A pyramid must have a square (all sides being equal) at its base, with lines drawn from four corners upward to meet at what is (or would be) a point (the apex). I have often heard people refer to Tetrahedrons as pyramids. In fact, the popular "pyramid" shaped tea bag is in fact a tetrahedron.
The five platonic solids
The tetrahedron is the first in a series of five shapes referred to as the platonic solids (after Plato). They are (in order) Tetrahedron, Cube (Hexahedron), Octahedron, Dodecahedron, and Icosahedron. The tetrahedron is a form made of four connected equilateral triangles. The five platonic solids are distinct in that all of their sides are the same shape.
Once again, the pyramid, in contrast to the tetrahedron, combines a single square and four triangular sides, equal in shape. The pyramid shape can be elongated or flattened and remains a pyramid as long as it remain three-dimensional. The only difference among pyramids, outside of the material they are made of, is the angles of its sides, dictating how steep the pyramid is.