붙다 is the intransitive one — the thing sticks by itself. The compounds here are 눌어붙다 (food catching and sticking to the pan), 엉겨붙다 (clumping or clinging together, used both of noodles and of three brothers piled on each other), and 옮겨붙다 (a fire spreading to the next building). It also covers something being attached to you or to a thing: a name that got stuck on it, or a 10% tariff added. Use 붙이다 when you are the one doing the sticking.