Plain version: a contract duty is one you agree to; a tort duty is one the law places on everyone. If a roofer skips a job he promised to do, he breaks a contract. If he drops shingles on a stranger walking by, he commits a tort: a civil wrong, apart from any contract, that a court can remedy with money damages.
A tort is not the same as a crime. The state prosecutes a crime; the injured person sues over a tort. One act can be both, but the injured person's lawsuit for damages is the tort side. Liability insurance responds when the insured is legally responsible for injury or damage to someone else, which is why P&C agents study tort law.
The CDI outline groups torts into three kinds:
Intentional torts: the wrongdoer chooses to do the act, such as libel, slander, false arrest or battery.
Negligence: harm caused by carelessness. California Civil Code §1714(a) makes everyone responsible not only for willful acts but also for injury caused by a want of ordinary care or skill.
Absolute or strict liability: responsibility without proof of fault, for certain activities or relationships.
The Casualty outline adds a fourth item to the same list: vicarious liability, where one party answers for another's wrong because of their relationship. It is explained on the gross negligence and vicarious liability page of this chapter.