If you’re trying to look up the term universal family, my recommendation is to not to use the search string “universal family” in Google.
Update. In the comments, Matt Heath points out that this post is already in the top ten for “universal family”, which means that I can be the change I want to see in the world. A universal family is basically a fine moduli space. A common example of a universal family is a classifying space in algebraic topology. A description of universal families in the context of elliptic curves can be found in Dan Edidin’s What Is… A Stack?