You've two choices. Pull it and walk skinless for the rest of your life, or cut it with some small scissor-like thingies you can find in your parent's bathroom. Nail clippers work, but ideally something with a slanted blade. Better yet curved.
Put tape over the edge of your skin where the hangnail is. Make sure it sticks and is tight. This will prevent it from pulling your skin with it. Duct tape and medical tape work the best. Then with some tweezers, pull the hangnail straight as if extending it out of your finger tip. It will hurt but it will be quick if you do it right. Antibiotics ointment on the hangnail hole. Bandaid.
Get one of these for hangnail, some nails are prone to have them and they will always come back. You could use a file to keep them short and smooth so that the hangnail has no chance to frizzle out and rip of the skin. That is for hangnails at the side that stem from the nail.
For the ripped skin at the root of the nail, keep your cuticle hydrated (they will automatically if you do skincare). Don't let them take over you nail. Again, that cuticle clipper will also help with rampant cuticle. It's horned skin that will eventually rip when it's dried out, which will lead to nasty:
It also helps to remove the dead skin on the nail after a shower (with your nails on the other hand). But if you have short nails or paper like nails you would need another instrument.