As @Noah Witherspoon correctly, says triangle subdivision does not work as well as quad subdivision. Although, in the beginning triangles could not be subdivided at all. However, he does not really explain why that is the case. Which is useful information and explains why quads are preferred and how to use them.
First, observe that a triangle does gets subdivided into 3 quads in many schemes. Since you now have a all quad mesh, clearly keeping the subdivision all quad is not in itself the requirement. There has to be a more profound reason than just being four sided.
Image 1: You can subdivide a triangle into 3 quadrangles
The reason lies in what has become called edge loops. The person doing the modeling have to anticipate how the subdivision happens as the subdivision is going to be the final shape. Unfortunately humans are only really good at deciphering the shape of the object along the edges of your primitives edges. By formulating the shape into continuous multi edge long loops helps us predict the shape after subdivision and importantly after deformation by bones etc.
A triangle has a nasty way of terminating the loop so we do not understand what happens with the shape within and out of that shape. The subdivided mesh thus has a tendency to behave uncontrollably, causing undesired bumps. Note: It is possible to subdivide triangles in a way that this does not happen, they are just harder to work with and working with quads were well known by then.
Now this is not actually the original reason, only it happened in roundabout way. The original reason what that the geometrical patches that they did use a as parametric primitives are square in shape. As extending a line into a surface naturally takes a square shape if you just extrude out. Having a triangle causes one edge to be degenerate and have a singularity. But this is very much related to the subdividing reason as it can be shown that a subdivision surface is just a general case of a spline patch.
Image 2: Original parametric surfaces were extensions of curves, not arbitrary meshes and these shapes naturally tend to be square.