I'm not sure how the term is being used here:
Let $R$ be a commutative ring and $X_1,\ldots, X_n$ indeterminates over $R$. Set $P = R[X_1, \ldots, X_n]$.
Given a ring homomorphism $\phi: R \rightarrow R'$ and $x_1, \ldots, x_n \in R'$, there is a unique ring homomorphism $\pi: P \rightarrow R'$ with $\pi\restriction_R = \phi$ and $\pi(X_i) = x_i$ for all $i=1,\ldots,n$. Another way to state this is that $P$ is a universal example of an $R$-algebra with $n$ distinguished elements.
How is it used in general? Also, this example was used as an example of a "universal mapping property" and could you help clarify what this means?
Thank you!