Both 还是and或者have meaning of choice. 还是 is used in a question sentence. Do you eat apple 还是 eat banana? Ask a question. 或者 is used in a declarative sentence. I eat apple 或者 eat banana. I can choose eat either apple or banana. 或者 is not used in a question sentence. 还是 also can be used in a declarative sentence, but means I don't know. I have no idea of she ate apple 还是 ate banana after lunch. I don't know she ate apple or ate banana after her lunch.

The same point between 还是 and 或者 is "or", in that case, 还是 is used when you ask and 或者is used when you state. For example: Which color do you like better, white or black?—你更喜欢什么颜色，白色还是黑色？ I'll go to Canada by sea or by air.—我将乘船或者坐飞机去加拿大。 The difference: 还是：1.still: It's still Mary win the match.—还是Mary赢得了比赛。2. had better: You'd better go home.—你还是回家吧。 或者：1. maybe: Maybe she is right.—或者她是对的。

Agree with Ricky, and what's more, 或者is more written, and 还是 is more spoken

every time a question was asked about difference, the first idea spings to our mind is there got be a difference, but in chinese, loads of words could be exchangable, admittedly different words have different explanation according to a dictionary, yet in pratice these differences have been ignored by most of us, and it dosen't matter,even chinese would not bother that. fluent and right pronoucation matter more than that. since i believe u r not going to work for 人民日报，so i suggest that: ignore it.

还是：1.（or），你去还是他去? you go or he？（？is the point） 2.(still),虽然整天下雨，他还是去上学，it was raining all the day,but he still went to school. 3.（had better）,你最好还是回家，you‘d better go home。 （give someone a suggestion that should take the best choice ，imply that it’s not a good choice but only one） 或者：1.（perhaps，maybe），或者（或许）他是对的。maybe he‘s right.（but the difference between “或者”and“可能” in this situation is difficult to tell.reading more is need ,i think) 2.(or),"谁跟你去？“”你或者他“ ，"who will go with you？""you or he."（"谁跟你去，他还是她？”"who will go with you，he or she？“） i dont think there are something in common between the two words. sorry for my english.