I was under the impression that a crime is always an offense against the state and never an offense against an individual. If it were an offense against an individual than it would be a violation of civil law rather than criminal law, wouldn’t it?
Civil cases are those in which the courts resolve a dispute between citizens, yes. However, if one person goes outside the law to achieve their means (assault, theft, rape, murder), then it is criminal law. Both perp and victim may have been civilians, but the police are paid to prevent those crimes.
If crime is not something you thought, what is a hate crime?
It’s an extra level of mens rea. It increases potential punishment for a crime. Being a bigot itself isn’t a hate crime.
Take murder, for example. If you intend to kill someone, and kill them, then that’s murder. If you intend to kill someone because they happen to be Asian, and you kill them, then that’s a hate crime murder. Society thinks that a hateful reason for committing a crime makes the crime even worse, and so the punishment is more severe.
See more here and here.