안녕하세요 is the standard greeting in Korean, equivalent to saying "Hello" in English. It would translate literally to something like "Are you well?" or "Are you peaceful?" so it is usually said with a slight uplift at the end indicating a question. It is also used as a farewell.

When used to say 'goodbye' there are some variations:

• 안녕히가세요 is used when speaking to someone who is leaving
• 안녕히계세요 is used when speaking to someone who is staying when you are leaving

When speaking to a child or a friend, you can just say 안녕 in all cases.

[edit] Grammar explanation

안녕하세요 is formed by combining 안녕, meaning 'peace', and the verb 하다, 'to do'. The 하다 is then conjugated in the honorific form to produce 하세요.