Both of these conjunctions can be translated as "or" in English, but they differ in usage: in general, 或者 is used in non-interrogative sentences to indicate a choice:
我妹妹想研究音樂或者文學。
還是, on the other hand, is more often than not reserved for alternative questions:
你上午去﹐還是下午去﹖
It can also be used in subordinate clauses with verbs such as 知道, 告訴 etc:
請告訴我這個人是老師還是學生﹖ 我不知道她是老師還是學生。
In cases in which both 或者 and 還是 can appear in questions, there are some subtle differences in their anticipations and tones:
1. A: 你想去中國還是日本﹖
B: 我想去中國。
2. A: 你想去中國或者日本嗎﹖
B: 我想去﹐我兩個地方都想去。
In dialogue 1, A asks an alternative question which anticipates a choice from the answer. The two terms with 還是 are mutually exclusive. In dialogue 2, the question is a general one using 嗎 at the end. It does not anticipate a strict choice from the answer, so B can give a general answer: 我想去 -- yes, I do -- and then add his more specific thought: 我兩個地方都想去. The two items with 或者 in the question are therefore not mutually exclusive.
Sometimes, however, 還是 can also be used in a non-interrogative sentence. For instance:
去看朋友﹐還是去看電影﹐他一時(??)不定主意。
Notice in the sentence the tone is still somehow interrogative, compelling 他 to make a choice. In contrast, 或者 in this case can only convey a sort of explanatory note:
你去﹐還是她來﹐你明天一定要作決定。 (interrogative)
或者你去﹐或者她來﹐沒有其他選擇。 (explanatory)