Fawn, as a noun, means a young deer.
The phrasal verb, to fawn over, means “to give care and attention to someone, the way a mother deer gives care and attention to her fawn—constantly and with much affection.”
Although this might sound like a good thing, the phrase has negative connotations when describing human behavior. It means too much attention and care. A person fawning over someone is seen as desperate to please the object of their affections, in order to curry favor—to make a person like you by using flattery.