The American two-party political system allows for an unlimited number of parties to participate. Anyone can establish a political party, and some minor parties, like the Libertarian Party or the Green Party, have a significant number of members and field candidates for election at every level. In practice, however, power is shared between two major parties, the Democrats and the Republicans. The president is inevitably either a Democrat or a Republican, despite the presence of independent candidates. There are a number of reasons for this, two of them being the advantages the party candidates enjoy in terms of media coverage and funding. Even Donald Trump, a billionaire with a high public profile, found it most effective and convenient to run as a Republican candidate. The Senate is made up almost entirely of Democrats and Republicans; there are currently two independent Senators, both of whom caucus with the Democrats. At the time of writing, every single voting member of the House of Representatives is either a Democrat or a Republican.
The United States has an unusually extreme version of the two-party system. Other countries, such as Great Britain, also have two major parties, but there are sometime opportunities to form coalitions and share power. In such circumstances, the boundary between a two-party system and a multi-party system is often unclear. The United States, however, essentially has the purest two-party system possible without restrictions on third parties. This is facilitated by a first-past-the-post system in elections for all major positions. Without proportional representation, third parties may garner a large number of votes without ever gaining power.
The two-party system means that a lot of the disputes in American politics take place within the major parties. Both major parties have several large factions, as well as numerous smaller ones, which often operate like parties within a party, attempting to push the public agenda in a particular direction.