If your eyes are bleary, they cannot see very clearly. You are tired, you have just woken up, or you have been crying, so the eyes look watery, heavy, and a little red, and what you look at is not sharp. The pairing you will meet again and again is bleary-eyed: a bleary-eyed student at seven in the morning. You can also say a bleary look or bleary vision. Though actually, people stretch it off the face a little — a bleary morning is a morning that feels like those eyes. It is descriptive, not slang, and not a medical term. It does not by itself mean drunk, even if a tired, hung-over face can look bleary too.