"What constitutes a family?" means "What makes up a family?" To constitute is to make up a whole from smaller parts, or constituents.
Parents, kids, pets, and other family members constitute most people's idea of a family. The Constitution tells what our country is all about and supposed to be made of — in other words, what constitutes the United States. This is word that can deal with the present or the future. The items listed in a recipe constitute what will be in a cake; later, the individual ingredients constitute the cake. Eleven players constitute a football team on the field. We can even say that abstract qualities are constituted. For example, some people think wearing sunglasses constitutes coolness. Since everything in the world is made up of other things, there's a lot of constituting going on.
