The US has three branches of government, the Executive, the Legislative, and the Judicial. We can ignore Judicial for this conversation.

The President, currently George Bush, is in charge of the Executive branch. He and it are responsible for the day-to-day running of the country, running the military, citizen benefits, collecting taxes, running foreign affairs, etc. The elections we just had didn't affect the Executive branch. (The president is somewhat equivalent to the British Prime Minister.)

These elections changed the makeup of the Legislative branch. The Legislative branch, or Congress, is in charge of making laws. These laws range from criminal offense laws to making budgets, declaring war, and so on. (Incidentally, the US has not officially declared war since WWII.) We have two subdivisions of Congress: the Senate and the House of Representatives. A proposed law must pass both divisions (or Houses) in order for it to become enacted. (Congress is somewhat equivalent to British Parliament, and the Senate and the House of Representatives somewhat equivalent to the House of Lords and the House of Commons, respectively.)

What's happened is that the Democrats have a majority in both houses of Congress, which means that it will be much easier for them to pass laws. Formerly, the Republicans had a majority in both houses. What this means to the President is that it will be harder for him to propose laws and get them passed. While the President has no lawmaking powers, it is in his interest to encourage Congress to pass laws that benefit, notionally, the country, but, in reality, his agenda. When the whole of Congress was controlled by his party, it was relatively easy for him to get laws passed, like the recent one that effectively gave him carte blanche to torture "enemy combatants" and declare anyone he wanted to as an "enemy combatant". Now that the opposing party is effectively in charge of making laws, he becomes hobbled in his ability to promote new laws.

It's also a signal to the Republican party that the US public is not happy with their performance.