I am not sure, if I understand you clear, so I really need some additional explanations what does each of the exams' names include. For example, "multiply choice" - is it about an exam when they give you questions with several varianst of answers marked a, b, c an so on, so you have to choose one of the answers? If so, then in Russian it is called "òåñò" [pronounced as test]. If it is something different, then please explain, so I can give you another answer.
As to "essay". If it means writing some text based on given exact topic to check up ability to express ideas about the subject, then usually we call it "ñî÷èíåíèå" [pronounced as sachinEnije].
As to "open book" - I have no idea what could be meaning of that, so here I expect describing of the exam to give you an answer.
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