Something that's broken is severely damaged — it's either split into pieces or doesn't work anymore. Your broken alarm clock is no help at all when you're trying to get to school on time!
If your arm is broken, the bone is physically damaged and needs time to heal before you can use it the way you normally do. And if your mom's favorite teapot is broken after being dropped on the floor, it may be beyond repair. Other things can be metaphorically broken, like a family split by divorce, speech that's not fluent, or your heart: "It's sad that you guys split up, but you're not going to die of a broken heart."