응원 is more of cheering someone or something 지지 is supporting a side, like a candidate of an election 성원 to me it's support and cheer from a wide range of people usually follows up with an appreciation, for example, "Thank you for your support (성원), because of it we could met our goal (of any kind) 지원 is more of an actually support rather than just a will. Like money support or government support. 뒷받침 literally means back-support of anything. Can be used as noun or verb. It has a bit of 'hidden' nuance to it, because of the word 'back'. So for example, My academic success would've been impossible if there was no support(뒷받침) from my colleagues.