A breach is a violation of a law, duty, or promise. If you'd contracted to mow your neighbor's lawn and don't do it, he can sue you for breach of contract. Or he can mow the lawn himself.
Breach traces back to the Old English verb: brecan, "to break," which is what breach really means — "a breaking of something." Breach of contract is a legal way of saying you broke a promise. If your great aunt Edna is a stickler for propriety, she may be horrified when you eat your meat with the salad fork, considering it a breach of good manners.