I refer to this paper on Moduli Spaces by Ravi Vakil.
I am uploading a screenshot:
What can possibly be a universal family over $G(k,n)$? For example, let us take the set of all linear subspaces of $\Bbb{C}^3$. What would the universal family or $U$ be here?
As written in the text, $U$ is called the tautological vector bundle over $G(k,n)$. Each element in $G(k,n)$ is a $k$-plane in $\mathbb{C}^n$. $U$ is the vector bundle on $G(k,n)$ consisting of all pairs $(P,x)$ where $P\in G(k,n)$ is a $k$-plane and $x\in P$ is a vector in that plane. The horizontal map is the inclusion and the vertical map maps $(P,x)\mapsto P$.
See also this Wikipedia article.