Hate crimes are prejudicial crimes against a person or persons in a specific group or class that the criminal somehow feels is inferior for its existence. This hate/prejudiced makes them think that it's somehow their responsibility to punish these people for being who they are. It could be against persons of color sexual orientation or religion.
Hate crimes are crimes motivated by hate towards a certain group. The best examples I can think of are the sensless violence toward gays (I'm one) and when Christianity nearly killed off paganism (but my people still survived).