I think pohnpei's response is right on. Ours has been designed as a two party system, first by the Founding Fathers in order to avoid what Washington called "The dangers of faction". Since that time, the two party system supports and continues itself, making it very difficult for other interested parties to enter the race.
We should acknowledge that there are many more than two parties in the US today. The Green Party, Socialist Party, Communist Party and Reform Party to name a few. These parties always put up candidates for the Presidency, and always lose badly. They are not even allowed to attend debates.
The Democrat and Republican parties have loads of cash, name recognition, a base of support, and a hammerlock on power. They also try to absorb other political movements and would-be parties by including some of the issues important to them in their party "platforms" adopted at the national conventions.
Since the War Between The States, one social system has held hedgemony in the U.S. That is big business. It started out as mercantile and industrial, often called the industrial social system; it is today corporate big business. Both the Republicaln party and the Democratic party serve its interests very well, hence they have survived and prospered.
See Post 6 under the discussion linked below.
Pohnpei397's answer above concerning single-member legislative districts is also a part of the reason.
I think that much of this depends on what you believe. From an optimistic point of view, I think that one could argue that the primary parties have demonstrated durability in recognizing the lessons of the Civil War. While as passionate of issues as slavery have emerged in the American political discourse since the Civil War, both parties have understood the need and reality for negotiation and lasting compromise. This partnership has allowed for effective rule over the last 150 years. On a more cynical level, perhaps the reason why both parties have demonstrated durability is to ensure that no other party or force takes their share of the market. Since the Civil War, America has been dominated by the two party system. This alliance of sorts has ensured a great deal of control over the political landscape and the trappings of power and wealth that accompany it. In the end, perhaps the durability demonstrated has been more out of political convenience and benefit than anything else.
In my opinion, this is largely because of our electoral system -- the rules for elections.
In our system, it is extremely hard for a third party to get going. We have these winner take all elections and single member districts where a party that gets 10% or even 25% of the vote will never ever get any representation. So if a party can't get close to 50% on a regular basis, it's never going to go anywhere.
Take the Tea Party types, for example. If they started their own party, they'd be nowhere. All that would probably happen is that they would help more Democrats win. So they have to try to work within the Republican Party.
Because of our rules, our two parties have just absorbed movements like that which could have potentially become third parties.