Well the roots of parties were based around our two competing ideas about government when the US was created. (Federalists vs. State's Rights.) Over time, more political ideals have gotten attached to both party platforms, and they've even exchanged some political philosophies in the past.
We have other parties in the US (The Libertarian Party, the Green Party, the Independent Party), but I don't think any of them have ever garnered more than 10% of the national vote. The Republican and Democrat parties appeal to broad swaths of the American public and their ideals. The independents tend to focus narrowly (and usually on the fringe) on certain issues like the economy, the environment, ect....And so they tend to run on only one or two of the big issues people care about. That's a platform with only two legs.
The Independent party is literally the party of "so fucking sick of this shit I'm an independent." You'd think the Independent Party would be stronger these days, but no. Without any platform other than "sick of this shit", no one can really vote for them except on a candidate by candidate basis. They're a party with one leg for a platform. And anyone with that good a chance to win never left the two dominant parties to begin with.
The 3rd parties also don't have these vast, national fund-raising and campaigning organizations to back them. They don't get a meaningful share of all the political contributions that businesses and voters give to the two major parties, they don't get the support or endorsement of serious celebrities or influential people. They can't afford the media blitzkrieg of ads that the other two parties can pay for. In terms of what's required to be relevant in American politics, the 3rd parties have almost nothing going for them, least of all the public's attention.
Voting for the "3rd parties" in America is usually considered throwing your vote away, because it's one less vote for either of the two parties at the end of the day.
We essentially have a parliamentary system where only two parties ever have a chance of succeeding. Which is why we get political deadlock so often. In parliamentary systems government is ruled by a coalition of multiple parties. It requires compromise for government to even function.
Our system tends more towards establishing dominant political control for one party, while the other takes an obstructionist position (and claims they're saving America by doing so, so they can get elected to Congress/Presidency/Governors of States in the next set of elections.) Compromise isn't necessary all the time (in fact only when Americans have elected a split House and/or Senate), and much of what goes on is motivated by increasing the political control and favor of one party and decreasing them for the other. It often has less to do with the actual legislation/topic/investigation/appointment/war/ect...under debate than it has to do with scoring points with voters for the next election. U.S. presidential elections aren't even until late 2012, and the U.S media has literally not shut up about them since late 2009 talking about who is going to run, and what something that happens today means for an election a fucking year from now, if that gives you any indication how dysfunctional we are politically. And if they're not harping on presidential elections (which Americans care a lot about, relatively speaking) early, there's the Congressional elections between every presidential election (which most Americans care a great deal less about.) It's a constant cycle of jockeying for advantage, and the issues are playing pieces.
And so we also get a good amount of factionalism within each party, usually more in the party that doesn't control government.