Something that's compromising makes you vulnerable to being embarrassed or incriminated in some way. If your teacher knows compromising information about you, you might want to ask someone else to write your college recommendation letter.
It's common for the adjective compromising to describe the noun situation. If your neighbor gets hit with a water balloon and looks up to see you leaning out of your window with another balloon in your hand, that's a compromising situation for you — it makes you look suspicious. Another way to use compromising is to mean "willing to compromise." It would be compromising of you to agree on pizza for dinner, even though you had your heart set on lasagna.