Cyclomatic number or excess are indeed common names for the quantity $|E|-|V|+1$ (as other answers mention). Let me add that the correct quantity to consider when you have $k$ components in your graph $G$ is $c(G):=|E|-|V|+k$. Then $c(G)=0$ means exactly that $G$ is a forest.
Also, the number $c(G)$ occurs in algebraic graph theory as the dimension of the cycle space of the graph $G$.