떼다 is separating one thing from another, usually by pulling it off a surface: peeling it off like tape, asking how you are supposed to get it off, pulling it off and planting it next to the others, and in a folktale the goblins promising to remove the lump without hurting a bit. The adverb is nearly always the word for like this, since someone is demonstrating. 붙이다 is the opposite, sticking something on.