One common behavior is extreme jealousy. For example, if his girlfriend just smiles at another guy, he might get very angry. Another is controlling her social life, like not allowing her to go to parties or hang out with friends.
Often, a possessive boyfriend will constantly check up on his girlfriend. He might call or text her multiple times a day just to make sure she is where she said she would be. Also, he may try to isolate her from her male friends. In a real story I heard, the boyfriend made his girlfriend cut off all contact with her male coworkers. He was afraid she would fall for one of them.
A possessive boyfriend in real stories may show possessive behavior by being overly critical of his girlfriend's appearance when she dresses up for an event without him. He might think she is trying to attract other men. He also may demand to know all of her passwords, whether it's for her phone, email or social media accounts, which is a huge invasion of privacy.