The Senate needs sixty votes to stop debate, but only a simple majority to pass legislation, with the exception of foreign treaties.
I am completely ignorant on this point, probably because it's never come up before. If we have 49 Republican Senators, and 49 Democratic Senators, and 2 Independent Senators (Sanders, Lieberman), how do the Democrats have a "majority." If I understand correctly, it is the "majority" party which can then name their members to specific offices, etc. Does a "majority" require 50 Senators. If not, isn't this kind of a mess?