Why does the USA only have two political parties?

First, as others have mentioned, there are more than just two parties in US politics. Well-known third parties include the Libertarian & the Green Parties. And there are some that are significant on the local level, such as the Vermont Progressive Party, which has members in both houses of the Vermont state legislature.

But the reason there aren't any significant third parties is simple: lack of infrastructure. By "infrastructure", I mean some form or organization that continues to exist even after a failed election.

I'm not very familiar with how elections work in countries other than the US, but in this country running for most offices is a very complex & labor-intensive endeavor. It involves much more than getting one's name on the ballot, knocking on doors, & participating in debates & meeting with people to gain their endors*m*nts. (In some races, there are no debates or organized groups who can offer useful endors*m*nts.) Knowing how to do this comes from experience, that is having people around with this experience -- or have participated in previous elections.

Now after a successful election is that those who were the most useful get hired on as staff -- which means they are available for the next election to other candidates of that party. But what happens with these people after an unsuccessful election?

When it comes to the Democratic & Republican parties, there are paying jobs that can be given to these people so their knowledge is not lost. Often they are jobs in the party, or other political appointments, or sometimes make-work positions in the private sector. These people comprise what you might call the party infrastructure.

But when it comes to an unsuccessful election with a third party, there is no easy way to keep ahold of them -- beyond appealing to their loyalty & ideology. Which works only so far: working fiercely 6 or 7 days a week to get someone elected, only to lose badly (most third parties get less than 10% of the vote), is discouraging. The Democratic & Republican parties are challenged to recruit volunteers to work campaigns, which means running a state-level third party is even a bigger one.

And then there are various legal barriers in place to discourage third parties; these vary from state to state. (On the other hand, it's an open secret that Republican operatives will help a Green candidate get on the ballot so to draw votes from the Democrats in a close election; & yes, the Democrats have been known to help a Libertarian candidate for the same reason.)

Infrastructure is the reason why Bernie Sanders, the sole Socialist currently in the Senate, decided to run as a Democrat: the Democratic party has the infrastructure that a third party candidate can only dream to have.

An important issue is that the American President is elected separately from the determination of the majority party in the legislative body in the United States.

In most European countries, the head of government is the leader of either the majority party in the legislative body (House of Commons, L'Assemblee, Bundestag, etc.), or more likely, the head of the majority coalition, in a multiparty setting. In this kind of situation, it makes sense for a number of small parties to form in order to represent competing interests in the legislature, then band together to choose the national leader. Under "European" rules, the President of the United States might be Nancy Pelosi, House Majority Leader, and we, like Germany, would have had a female CEO already.

But in the U.S., the race for the Presidency is distinct from the legislative election process discussed above. It has its own special rules, including the Electoral College alluded to in the last paragraph, which cause the Presidential race to lend itself better to a two party system. Basically, a President needs to win with a plurality in each state, and then win a majority of state electoral votes. In 1860, Abraham Lincoln won with about 60% of 60%, that is a majority of votes in states with a majority of electoral votes. His theoretical vote total could have been 36% or even less, but he won a few "stray" votes elsewhere to come to 40%.

"Third party" candidates (Ross Perot in 1992, Robert LaFollette in 1920) tend not to gain traction. (One partial exception, the "Bull Moose" Party, was led by a former President against the incumbent President.) Although this is true mainly at the Presidential level, this has also discouraged fragmentation of the legislature as well.

Here's how "European" rules might have worked in an American context. In the 2000 Presidential election, George Bush Jr. and Albert Gore Jr. both won 48% of the popular vote. Ralph Nader, of the Green Party won 3%. Assuming that legislative seats had been apportioned in proportion to the popular vote, Nader, with his 3%, would have formed a coalition with either Bush or Gore (probably the latter), to arrive at 51%. Conceivably a winning "bidder" (possibly Bush), might have made Nader president in exchange for the Vice-Presidency and a dominant Cabinet. (In 1933 Germany, conservatives made Adolf Hitler Chancellor in exchange for the Vice-Chancellorship and 8 out of 11 Cabinet seats; they thought they could control Hitler, and they were wrong.)

In 2000, nothing of the sort happened. In fact, Bush was made President by the electoral college even though he had fewer popular votes than Gore. Not only did he have fewer votes than Gore plus Nader, he even had fewer votes than Gore and Nader together in Florida (the "tipping point" state).

You are missing the trees for the forest.

Congress and the Presidency are two very different beasts, elected in two very, very different ways.

The dominance of two political parties in presidential races is because all of the larger and medium sized states are winner take all at the Electoral College. Many of these, the so-called swing states, will be decided by quite small margins. Running as a third party candidate ensures the victory of the Party furthest from your own political views, as most recently happened with Ralph Nader's run in 2000. Any habit of making political enemies in this way out of potential allies is a fast road to nowhere.

By contrast, in Congress one must really acknowledge that the lack of party discipline in both houses means there is in reality 535 political parties: 435 in the House and another 100 in the Senate. Yes, almost all of these are associated with one or other of the two large funding organizations known as The Democratic Party and The Republican Party - but these are at best broad coalitions of general political alignment. The vote of each Representative and Senator is effectively for sale to the highest bidder on every call of the roll. That's why there is so much, and has always been so much, pork barrel in American political bills since urbanization of the States started for earnest about the same time as post-Civil War Reconstruction.

Finally, what the drys and suffragettes discovered in the late 19th century is that a far more effective means of influencing political decision-making (than running third party candidates) was to, for lack of a more descriptive term, "extort" compliance from already elected Representatives and Senators. A vocal minority of voters determined to block-vote on a single issues easily swings elections, and more and more easily as overall participation drops as it always does on off-presidential years. This in turn was possible because of the lack of Party Discipline and any true Parliamentary Whip as in a Westminster system.