to put on clothing; to dress someone or get dressed oneself, and by extension to take something on as if slipping into a garment.
At its root this is the ordinary word for getting dressed. It covers both directions of the same act: putting a garment on another person, and putting clothes on oneself. In its settled forms it can also describe the resulting state, simply being dressed in something or having it on.
From that plain picture the New Testament draws one of its favorite images. Just as a person deliberately puts on a coat, so a believer is told to put on Christ, to put on the new self, to put on compassion, or to be clothed with power. The point of the picture is that the change is chosen and worn openly, the way clothing is both a decision and something everyone can see.
The two uses are not really separate words but one continuous idea. The metaphor stays close to the wardrobe it comes from, which is why English keeps the same phrasing throughout: you put on a robe, and you put on Christ, with the same verb doing both jobs.