When you ask someone to pick one from two choices, use 还是:
Are you a boy or a girl?
Do you drink tea or coffee?
When it's maybe one or the other or both or neither, use 或者:
I'm going to Beijing or Shanghai next week.
Today or tomorrow, we're going to meet up.
When used with phrases like "I don't know...", use 还是:
I don't know if this is his or hers?
I'm not sure if he's coming this morning or this afternoon.
还是 has other meanings too like when used with 吧 for "had better":
We're going to be out all day. We'd better take something to eat.
He needs the money today. You'd better go to the bank.
When there are two options, you can use 或者 before both (one for "either" and one for "or"):
Either you ask him or I will.
Stop waffling. Either tell me or shut up.