As the first post states, the idea that tax cuts can stimulate the economy is Reagonomics or Trickle-down-economics. It's also called Supply-side Economics. While wealthy people would like to believe it works, since it means more money for them, it has never been proven. The idea is that if you give the wealthy money, they will invest in more jobs. Unfortunately, if you give the wealthy money they will put it in offshore accounts and find other ways to evade taxes. Give a poor or middle-class person money and he or she will go out and spend it. Give a wealthy person money and he or she will put it in the bank to earn interest.
It seems that most tax cuts have more influence on the businesses and wealthier people of the nation. A tax cut for middle class and lower class wage earners may encourage some more spending but for the most part it is not going to have enough of an effect to stimulate the economy in itself.
Also keep in mind that there are many kinds of tax cuts: income tax cuts, capital gains tax cuts, sales tax cuts, etc., each of which can have a different effect.
Pohnpei has explained well the economics that often motivates tax cut supporters, but also consider what a cut in the capital gains tax can do. This gives an incentive to investors, especially large investors, to re-enter the stock markets or real estate sectors, or to invest more heavily in them because there is more money to be made. This stimulates the markets, adding wealth and encouraging businesses to spend and expand. All of these things have a stimulating effect on the economy. A sales tax cut, on the other hand, or sales tax holiday, can stimulate the purchase of big ticket items like cars and boats and other durable goods.
There are two ways to look at this -- a Keynesian view and a supply-side view.
By the Keynesian view, tax cuts work on aggregate demand to stimulate the economy. As the government taxes less, people have more money to spend. As this happens, aggregate demand increases and Real GDP goes up. This revives the economy.
By supply-side logic, however, tax cuts stimulate aggregate supply. As people get to keep more of their earnings, they work more (and companies produce more). This boosts aggregate supply.
These are the two theories on how tax cuts can help revive an economy.