"Здравствуйте" is more polite form of "зравствуй". Difference is approximately the same as between 您好 and 你好. Also "здравствуйте" can be seen as a plural form "зравствуй" for greeting group of people. "Здрасте" is a colloquial form of "здравствуйте", very informal. People who know each other well would most often say "привет" when see each other.

In Russian, we have two translations for the word "you". First option is "вы". It's a formal appeal to anyone. In this option we use "Здравствуйте". Second translation for "you" is "ты". It's appeal to close friend. In this option we use "здравствуй" or "привет" or "здрасьте"(in English it's a analogue of words "hello" or "hi")

здраствуйте is a correct polite form of greeting while здасьте is a short form. the latter is used by those who are too lazy to articulate properly hence it can seem a bit rude in a formal context. Also in talking the two words can sort of merge in something in between as "здарст[unclear short sound instead of "вуй"]те" In common I use this something in between

Здравствуй(те) - formal form Приветствую (вас\тебя) - hello my friend (I am greeting you) Привет - common (usually it is used between friends) Iformal (to chat with friends, for instance) : прет ('иве' is left out), здрасьте, здорова, приветик. Use of these words depends from situation and culture of person really =)

=)