If you are broken-hearted, the sadness is not a small one. Someone you love has died, or walked away, and it feels as if the middle of you has cracked. People also stretch it a little: a child can be broken-hearted over a lost dog, and a fan can joke they are broken-hearted that a band has split up. The joke only works because the real meaning is so heavy. You will also see heartbroken, one word, no hyphen. Same feeling. Broken-hearted is a little more storybook, a little more like something you would read in a song. In everyday talk, heartbroken is more common. Both are fine.