Sometimes the name subgraph matching is also used for the same problem. This name puts emphasis on finding such a subgraph as opposed to the bare decision problem.
The proof of subgraph isomorphism being NP-complete is simple and based on reduction of the clique problem. If subgraph isomorphism were polynomial, you could use it to solve the clique problem in polynomial time. Let n be the number of edges in G: you can then run the subgraph isomorphism n-2 times (with G1 being a clique of size 3 to n, and G2 being G) to find the largest clique in G.
Subgraph isomorphism is a generalization of the potentially easier graph isomorphism problem; if the graph isomorphism problem is NP-complete, the polynomial hierarchy collapses, so it is suspected not to be.
Also of main importance to graph grammars.