없어 means just "There's not...". 필요 없어 = "There's no need (for it or you to do)". As for "필요 안 해", it's because 필요하다 is an adjective while "운동하다 or 공부하다" is a verb - so you can say 운동 안 해 or 공부 안 해. If you do 운동, you'd jog or go to a fitness center. If you do 공부, you'd read a textbook and take notes. But what do you do to do 필요? Nothing! It's something that just happens. [이해하지 않아] or [이해 안 해] means - unlike in English - "I don't try to understand" or "I won't accept it." So we say "이해가 안 돼"(Literally, "It's not understood by me") or "이해가 안 가"(Well, just to make sense of it, I'd say it is meant to be "The process of understanding doesn't go(work)")

없어 is 'Be NOT' and 안해(아니하다) is 'BE NOT and Do Not' in english there are Two meaning of 'BE', 1st one is exist 2nd is state or condition... 없어 is firstone 아니하다 is second one. and 안해 --> 아니하다.. and has two meaning. 1) do not --> i don't do it (내가 안한다.) 2) Be Not --> It's not beautiful(이쁘지 않다.) 않다 is sorten form of 아니하다 so you might think ' I don't need --> 필요안해' NEED can be Noun and verb in english, but '필요' is noun in korean so it should be like 'there isn't any need' so you can't use don't(아니하다) with '필요' 이해하다 is understand, and 이해 is understading 생각하다 is think and 생각 is thinking.. these word is form Verb originally.. so you have to use 'don't' (아니하다, 아니되다) an antonym of 생각하다 is 생각 안하다 an antonym of 이해하다 is 이해안되다 when you use 이해 for knowlage things.. ex) I don't understand it 난 그게 이해가 안돼! an antonym of 이해하다 is 이해하지 못 하다 when you use 이해 for considering, figure out (other's situation) ex) I didn`t understand what was going on! 나는 무슨 상황인지 이해하지 못 했다. ---------------------------------- sorry this is all i can explain so far.. =,=;; pretty hard question for me.. I bet some others can help you ^^