The New Deal was first and foremost about resolving the economic crisis of the Great Depression. This had many components, however, including deflation (the increasing value of currency), unemployment, poverty, and confidence in the United States. Of course, the administration had the additional aim of preventing future disasters.
The New Deal aimed to increase employment through public works projects. This is what economists call fiscal policy--when the government attempts to stimulate the economy by spending money. According to many experts, such fiscal policies provide a very powerful benefit to the economy, especially in the short term. Hoover had perhaps been too risk averse to attempt employing so many Americans through government spending.
In order to address the issue of poverty, FDR introduced policies such as social security and unemployment benefits, which created a social safety net for people affected by the crisis. This was partly to help Americans, but it was also to alleviate the widespread sense of panic consuming the nation. Low confidence in government can cause issues with bonds, stocks, and other finical markets. A nation with such unrest cannot function economically. Perception is important in an economy, and it's difficult to feel like things are improving if poverty is so common. In this way, the goals of these programs were both to help Americans and to improve their confidence in government.
Perhaps what is less often discussed regarding the New Deal is the monetary policies that FDR's administration also took to help alleviate the crisis. There is a debate as to how important monetary policy might have been in the New Deal. One reason many historians focus on fiscal policy is because after the failures of the Hoover administration it was clear that monetary policy alone would not solve the issue; both were important. The New Deal had to make capital (fancy word for money) available for people looking to take loans or start businesses. With deflation so high, one way to solve this problem was to take the United States off of the gold standard, so that the cost of money could go down more easily. This also allowed the government to lower interest rates.
Changes such as Glass-Steagel and tighter government regulation were intended to prevent another economic disaster. This was another goal of the New Deal, but I won't go into too much detail here, since others have already done it so well.