My facourite example is that the integers $\mathbb Z$ may be obtained from a pushout diagram

$$ \begin{matrix} \{0,1\} & \to & \{ 0\} \\ \downarrow && \downarrow \\ \mathcal I & \to & \mathbb Z \end{matrix}$$ in the category of groupoids where $\mathcal I$ is the groupoid with two objects $0,1$ and non identity arrows $\iota:0 \to 1, \iota^{-1}:1 \to 0$ This pushout also nicely models the way the circle $S^1$ is obtained from the unit interval $[0,1]$ by identifying $0$ and $1$. Thus we see one value of the notion of colimit, of which pushout is a special case, namely to compare concepts in different categories. Also, we know most things internally about the groupoid $\mathcal I$, whereas the integers are infinite.