Short answer: Anything that violates the NAP.
Long answer: A crime is any aggressive action. For example, theft, assault, fraud. The NAP says that no one may initiate aggression against another; thus, any such action violates the NAP.
Note that this does not mean that all actions that avoid violating the NAP are morally correct. They aren’t. Many actions violate the Golden Rule, and are therefore morally improper, but that doesn’t mean that one can use violence in response to them, as that itself would be a NAP violation.
Of course it must also be noted that this is not the legal definition of a crime. All that means is that we do not live in anarchy. Under anarchy, that would be the definition of a crime, because anarchy does not authorize the violation of the NAP by anyone at any time. Using violence to prohibit actions that are not NAP violations is in itself a NAP violation, which cannot be considered legitimate in anarchy.