If you break a contract, it's a breach. If you're talking about pantaloons or feet-first babies, use breech with a double "e."
A breach is a break — whether it's a breach in contract or a breach in a dam. Lawyers like it because they can help if there's a breach of the law. Whales break free from the water when they breach, but no one wants them to breach the side of their boat because that boat would sink. Just ask the Pequod. Other examples:
"It is impossible to tell if there is a specific breach of any conflict of interest laws." (New York Times)
"All 68 passengers and crew are safe, authorities said, and the vessel itself incurred only a small breach in its hull." (Time)
A breech, double "e," is an opening at back of the gun. It's also the word for hindquarters. The breech is on the butt of a gun, but it's also sometimes the buttocks of a human. Seriously. Breeches is an old word for pants, or britches, the kind George Washington rocked. You can still get breeches, though, for horseback riding. If a baby is born feet- or bottom-first, that's a breech birth. Examples:
The snow clung to their boots and breeches, and the wind rattled the pines and made their cloaks snap and swirl. (A Dance with Dragons)
"Goforth said the first baby came out and then the second baby came out breech birth, meaning the baby was delivered feet first." (Washington Times)
Use breach is there's a break, but save breech for pants and babies.