1. To fail at something, to do something wrong/incorrectly (sometimes repeatedly). Someone that screws up a lot is called a "screw-up"
2. To describe hurting someone emotionally or physically. Commonly used when threatening the one you want to hurt.
Person 1: "I stayed up late last night to study but I couldn't concentrate that well."
Person 2: "You're gonna screw up that test."
"This is the 5th time in a row I'm getting a bad grade. I'm really screwing up."
"I couldn't stop stuttering when introducing myself to him. I screwed up so badly."
"Oh yeah!? You wanna mess with me, kid!? Ohh, I'm gonna SCREW YOU UP!"
Person 1: "Haha, I totally screwed that kid up, did you see his black eye?"
Person 2: "Yeah, he deserved it!"