Problems that are easy on bipartite but hard on general graphs
There are several well-known NP-complete problems that become solvable in polynomial-time for bipartite graphs. For example, 3-coloring is easy as bipartite graphs are precisely the 2-colorable graphs. Another example is independent set which is made easy by König's theorem.

Wikipedia also lists a problem that is NP-hard for general graphs but is trivially in P for bipartite graphs: the odd cycle transversal problem.