I am a U.S. citizen living in Russia with a multi-entry student visa. I noticed that on my migration card it lists the Russian Federation as well as the Republic of Belarus. Do I need to get another visa if I want to visit Belarus, or is what I already have sufficient?
In short the answer is yes you do need a separate visa for Belarus. Your Russian Student Visa doesn't allow you to enter Belarus.
The reason that it has the names of both countries is because the form they use is exactly the same. From the US Embassy in Moscow: