Район = village, neighbourhood Регион = region Округ = district Край = border, edge, periphery (of a town)

"Район" is a group of settlements, which counts as a small administrative and territorial unit in Russia. "Регион" can be defined as a "район" that covers a large area and has its geographical boundaries. In "регион" there is also a certain economical, social and cultural system of interconnection. "Округ" can be similar to "район" if we speak about a small administrative unit (for example, there are 10 of them in Moscow) and to "регион" if we speak about the whole country. "Край" is a huge administrative and territorial unit (there are only 9 of them in Russia as far as I know). Nowadays "край" as an administrative unit doesn't differ from "область" and it got its name from those outlying districts of the country. So it’s something like that ^_^

Every word from your list means a kind of formal division of Russian territory. The motives of why one city has become a part of one 'район' ('край', 'регион' or 'округ') and not the other, could be economic, political, historical or it could be no reason at all. If you need to find out 'район' etc. of some city, you have to check it in the classificators. PS Word 'район' could be also used to describe an area of any size. It could be an area contains a few houses, or a half of a city. You can understand what size meant from the context of speaking.

Попробуте понять, переведя эти предложения: В городе построили новый район (part of the city). Административным центром Кировского района является город Кировск. В большинстве районов Земного шара наводнения вызываются продолжительными, интенсивными дождями и ливнями в результате прохождения циклонов. В район бедствия вылетела группа врачей. Посмотрите карту Поволжского региона (часть страны Россия). Обама посетит страны Азиатского региона ( страны Азии) Горожане пришли в избирательный округ на выборы депутата. Первыми на помощь пришли солдаты Кавказского военного округа. Город Ноябрьск находится в Ямало-Ненецком автономном округе. Многим людям нравится отдыхать в Краснодарском крае.

Каковы различия между районОМ, регионОМ, округОМ и краЕМ?