To gift-wrap something is to put pretty paper around a present so the person cannot see what is inside until they tear it open. Shops do this a lot: you buy something, and they offer to gift-wrap it, often with paper, ribbon, and a bow, sometimes for a small extra price, sometimes for free. You can gift-wrap at home too. The paper itself is usually gift wrap (two words, a noun); the action is gift-wrap (a verb). Though actually, many shops now drop things into a bag with tissue paper instead of wrapping every box, and they may still call that gift-wrapping. Ordinary, friendly English — not fancy, not slang. You will hear it most in shops in the weeks before a holiday.