What is an independent, nonprofit group that receives and disburses funds to influence election campaigns called?
The best answer to this question is to say that an independent, nonprofit group that receives and disburses funds to influence election campaigns is called a 527 committee. This sort of group is named after a part of the United States tax code, Section 527, that defines what it can and cannot do while keeping its tax-exempt status.
The important thing to remember about 527 committees is that they cannot directly call for the election or defeat of any political party or candidate. In other words, 527 committees cannot fund advertisements (for example) that say "Vote Scott Walker for President” or “Defeat Scott Walker.” Instead, these committees have to appear to be educating people on particular issues. A pro-Scott Walker 527 might make ads that argue that anti-union activities like Walker’s are good ideas, but it cannot directly encourage people to vote for Walker.