To "give someone the third degree" means to interrogate them (ask them many questions, like an investigator or police officer might do) and it implies that you are rough, loud, persistent, and probably angry.
This phrase might be used by a teenager who comes home after curfew, and whose parents start questioning her where she was and what she was doing. She might yell back "I was just at the movies with friends! Don't give me the third degree!" to mean "I'm innocent, you don't have to ask me these probing questions!"
(In the legal system, offences are divided by how bad they are - first degree is severe, second degree is medium, and third degree is mild. The idiom "to give someone the third degree" is only used for third degree, not for first or second.)