어쩌다(the base form) is short for 어찌하다 ("acting how" or "doing what" in a questions, or "somehow" otherwise).
- 어쩌다 = short for 어쩌다가 = doing what. Asks the cause of some outcome.
- 어쩌면 = 1) if (you/they) do what (in a question), 2) perhaps, possibly (non-question).
- 어쩌지 = wonder what I/we should do. (-지 is a mild question ending looking for confirmation).
- 어쩐지 = 1) for some reason (that I can't pinpoint), 2) as I suspected.
- 어쩌자고 = trying to do what; with what intention.
- 어쩌다(가) 이랬어? How did you end up doing this = What led you to do this?
- 어쩌면 이 문제를 해결할 수 있을까? What could I do to take care of this problem?
- 그는 어쩌면 찬성할지도 모른다 = He might possibly agree for all we know.
- 돈을 잃어버렸으니 이제 어쩌지? = What do I do now that I lost the money?
- 친구는 오늘 어쩐지 우울해 보인다 = My friend looks sad for some reason today.
- 어쩐지 나를 자꾸 피하더니 이유가 있었다 = Somehow he kept avoiding me and there was a reason.
- 어쩌자고 직장을 그만두겠다는 거냐? = What are you thinking that you'll quit your job.