Both 還是(simplified form:还是) and 或者 can mean or, and I think the only different is that, "还是" is oral, and “或者" is formal.
You could just use ”或" to mean "or", you could also use “抑(yi4)或",but this word are used more in literature and you won't use it in ordinary life.
Thanks for Cocowalla's reminder.
I think you can use either of them in both question and statement sentence. There is only one thing to take care of. When you ask some to pick one from your choices in a simple sentence(no clause) , only ”还是" is acceptable, and when you express your choice in a simple sentence(no caluse), only "或者" is acceptable.
See my comments please.