What is a universal property?

Sorry, but I do not understand the formal definition of "universal property" as given at Wikipedia. To make the following summary more readable I do equate "universal" with "initial" and omit the tedious details concerning duality. Suppose that $U: D \to C$ is a functor from a category $D$ to a category $C$, and let $X$ be an object of $C$. A universal morphism from $X$ to $U$ [...] consists of a