НА НЕДЕЛЮ / НЕДЕЛЮ - for a week / a week - We can say about our plans using this phrase or tell people about our experience. These structures show us HOW LONG we have been doing anything. Or for HOW LONG we plan to do it. It's absolutely unimportant what exactly we do, as well as a result. It's only important for HOW LONG, how many days/nights/years.
But there is a small grammar difference. НА НЕДЕЛЮ (НА + Gen.case) - for a week - we use it after the verbs of motion. For example: Я поеду в Прагу на неделю; я летал в Лондон на 3 дня - I will go to Prague for a week; I went to London for 3 days.
Possible questions: НА СКОЛЬКО ТЫ ПОЕДЕШЬ? НА СКОЛЬКО ТЫ ЛЕТАЛ? НАДОЛГО? - For how long will you go there? For how long have you been there? For a long time?
НЕДЕЛЮ (Gen.case) - we use after the verb "быть" (to be). For example: Маша была там неделю, Оля будет жить в Германии три года - Masha was there for a week, Olya will live in Germany for three years.
Possible questions: СКОЛЬКО ТЫ БЫЛА ТАМ? СКОЛЬКО ТЫ БУДЕШЬ ЖИТЬ ТАМ? КАК ДОЛГО? - For how long have you been there? For how long will you live there? For how long?
By the way, when we speak about the present tense, we use only the second option: Gen.c. (НЕДЕЛЮ). For example: Я читаю книгу неделю (я ещё не закончила, процесс сейчас) - I've been reading a book for a week (I'm not finished yet, it's a current process); Я живу здесь 4 года - I've been living here for 4 years.
ЗА НЕДЕЛЮ - in a week - We say this way, when we want to express HOW QUICKLY we can do our work or HOW QUICKLY we've done the work. ЗА + Gen.case. - is a deadline, term. When this term ends, work is also completely finished. The result is only important. For example: Я прочитал "Войну и Мир" за неделю (это мой результат, я молодец, вау!) - I have read "War and peace" in a week (this is my result, well done, wow!); Я сделаю презентацию за день (я обещаю, я молодец, я закончу 100%, это будет мой результат) - I will make a presentation in a day (I promise, I'll complete the task by any means, it will be my result).
Possible questions: ЗА СКОЛЬКО ТЫ СДЕЛАЛ ЭТО? ЗА СКОЛЬКО ТЫ СДЕЛАЕШЬ ЭТО? - For how long have you done it? For how long will you do it?
ЧЕРЕЗ НЕДЕЛЮ - in a week - We use it, when we want to say WHEN we'll start doing something. We show how many days we will not do it. For example, the phrase "Мне надо позвонить директору через 2 дня, 29.09" (I should call my boss in two days, 29.09) means that today is 26.09.
"Я поеду в отпуск через две недели" (I'll go on vacation in two weeks) means that you'll work during the following 14 days and then will go on vacation. Possible questions: ЧЕРЕЗ СКОЛЬКО ДНЕЙ НАДО ПОЗВОНИТЬ ДИРЕКТОРУ? КОГДА ВЫ ПОЕДЕТЕ В ОТПУСК? - When (in how many days) should you call your boss?