* 빨리 - quickly, fast (can mean both quick response and high rate of speed). Most common word in the group.
* 얼른 - immediately; without delay. A colloquial everyday word for doing something quickly.
*어서 - 1) similar to 얼른, 2) a word of an encouragement or demand ("어서 오세요." = welcome.)
* 즉시 - immediately (즉=immediate, 시=time). More formal and urgent sounding than 얼른 or 어서.
* 바로 - right away; without delay, waiting, going astray, etc. (It is not always time-related.)
Only 빨리(빠르다 = fast) and 바로 (바르다 = right, straight, upright) have an adjective form.
바로 is not as much about quick action as it's about not straying or dawdling.
- 학교 끝나면 바로 집으로 와 = Come home straight after school (without hanging around or going somewhere.).
빨리, 얼른, 어서 can all be used to prompt someone to action in everyday situation.
- 점심 빨리/얼른/어서 먹고 나가서 놀자 = Let's finish lunch quickly and go out and play.
즉시 (and also 즉각) is similar to 바로 but sounds more serious and urgent
- 그가 돌아오는 대로 즉시 떠나야 한다 = We have to leave as soon as he comes back.