What are the different government parties?

What American laws give political parties the influence they have over elections and government?

• Political parties aren't mentioned in the Constitution, so - as private organizations - what is the legal source of their standing?

• Answer:

  tl;dr: The First Amendment There are a few laws that give political parties advantages over non-organized groups.  They get automatic access to ballots, and the states often use the same election machinery to hold party primary elections. But the party influence doesn't derive from that.  It's really the other way around: these election apparatus are acknowledging the influence of the parties.  The primary elections are a recent innovation, dating back to the 1960s, a century after the parties had settled into their present form.  They had held nearly all major offices ever since.  The parties could gather up the signatures they need to appear on a ballot so easily that actually making them do it would just be a hassle. The influence of the parties derives from something more basic: safety in numbers.  A group of people who agree to work together can achieve more than an individual can.  The larger the group, the more effective it is.  The cost is that the more people, the harder it is for them to agree.  Some split, and to be effective they have to organize their own group.  The net effect: exactly two parties.  Mathematically, it's called Duverger's Law, which is far more effective than any mere legislation.  Some laws do spring up to acknowledge the fact.  For example, the President and Vice President are elected together, effectively enshrining the notion of a party right into the Constitution without explicitly using the term.  The rules of the House and Senate aren't laws, but they also explicitly acknowledge the fact that there will be a majority and minority party. State legislative boundaries are gerrymandered with the tacit-but-obvious intent of enshrining one political party, but it's really more about defeat of the other party rather than an attempt to exclude additional parties.  The third parties are, on the whole, self-defeating without any legislation. Legislation has been proposed to try to combat Duverger's Law and give third parties and individuals a stronger chance.  Most of this legislation runs afoul of the right to assemble.  That right to assemble is the root of party power: take away the right to assemble and you won't be able to have parties.  I think very few people find this a good solution.