Demand-side economics (also known as Keynesian economics) is related to the federal budget deficit because it often prescribes deficit spending.
According to demand-side economics, the way to get out of a recession is for the government to spend more money. This gives people more money with which to buy goods and services, thus increasing aggregate demand. This is often called "stimulus spending," and was the core of President Obama's strategy for bringing the US out of the "Great Recession."
Of course, this sort of spending can lead to large deficits in the federal budget. Demand-side economists would argue that this is a small price to pay for improvement in the economy.