If you get punched in the arm hard enough, you will develop a hematoma (bruise) localized to the area of trauma. So why is it that you can get a black eye after breaking your nose?
When you break your nose, you damage adjacent blood vessels that release blood into the surrounding area. This is the same mechanism that results in a localized hematoma. However, the skin on your face has a looser layer of connective tissue underneath, which makes it softer than your skin elsewhere. The blood released from your broken nose can flow more easily through this loose connective tissue and often pools beneath the eyes. This blood then oxidizes, darkening and resulting in a black eye.