When you do something well, you do it in a good or satisfactory way. You can say, "I did really well on my French test," or "Finally, I slept well last night."
When well is a noun, it means "a deep hole full of water or oil." When well is an adverb, it describes the way something's done. If you're not sure when to use well and when to use good, think about what you're describing. If it's a thing, then you can call it good, like a "good book." If you're talking about an action, you should use well: "I always do well on math quizzes," or "You speak Chinese really well."