American liberalism, the dominant Ideology of the Democratic Party, was formed from two strands:
Classical liberalism, the philosophy formed in the Enlightenment by thinkers like John Locke, Voltaire, and Rousseau, and providing the driving force for both the American Revolution and the French Revolution, which states that the autonomy of the individual should be maximized, and the individual should be freed from whatever institutions are preventing them from reaching their potential, be it the Church or the State.
Progressivism, a populist reform movement in the early twentieth century, espoused by Presidents like Theodore Roosevelt and Woodrow Wilson, and grounded in Protestant moralism, which sought to make government both more responsive to the plight of the people, for instance using Constitutional amendments to deal with social problems like alcoholism and using government force to quash monopolies, and at the same time more representative of the will of the electorate, for instance instituting more democracy like the direct election of Senators and ending the corruption of Machine politics in the cities.
These two strands were merged into the ideology we call American liberalism by John Dewey and his followers, who argued that we needed a broader conception of liberty than the one maintained by laissez-faire negative-rights libertarians. The key idea can be summed up in a quote from Anatole France: "In its majestic equality, the law forbids rich and poor alike to sleep under bridges, beg in the streets, and steal loaves of bread." Basically, the idea is that the freedom to starve because you have no food is not a meaningful freedom at all, because it does not maximize your autonomy or allow your to realize your potential, which were important goals in classical liberalism.
Thus Dewey argued that we should recognize positive liberty as well as negative liberty, meaning that e.g. just as we ought to recognize a right to live without someone killing you, we similarly ought to recognize a right to live without dying due to lack of food. Thus American liberalism advocates that the government should play some role in the economy in order to give people autonomy and enable them to pursue their own happiness, along the lines of the "responsiveness" part of the Progressive philosophy. Thus Americans liberals still try to achieve the goals of classical liberalism, but they sometimes do it through Progressive means. (This is akin to how, as I discuss here, American conservatism tries to achieve the goals of traditionalism, but often through libertarian means.)
There's one other way that American liberalism differs from classical liberalism: classical liberals took a deontological perspective on liberty, viewing personal autonomy and the pursuit of happiness as things that are inherently worthy of being promoted, regardless of what they lead to. American liberalism, on the other hand, because it emerged partly from Progressivism, tends to take a more utilitarian perspective on such things, viewing autonomy merely as a means to an end, the end being increasing the happiness of as many people as possible. The liberal understanding of utilitarianism is perhaps best understood through the work of John Rawls, who proposed a thought experiment along roughly these lines: suppose that you're a soul waiting to be born, but you don't know which body you're going to be born into and what experiences that body is going to have. But before you're born, you have the opportunity to design how society and government should be. How should you design it in order to maximize your chances of having a good life? Well, you don't know what desires you're going to have, but whatever they are, you're probably going to want them fulfilled, so you would want a society that as much as possible allows people to pursue their own happiness. And you may end up being born in utter poverty that may prevent you from being happy, or you may start off being wealthy but then you may suffer calamities like disease and natural disasters, so assuming you're risk averse (as humans tend to be in most circumstances), you'd want the government to have a safety net to shelter you from such risks. So that's where the basic contours of the liberal position emerge.
Finally, let me mention something about foreign policy. The Progressive presidents advocated a very interventionist foreign policy, since they were motivated by the desire to help people as much as possible, even people abroad. Liberals still share some of this impulse, and are willing to support limited American military intervention in circumstances of extreme humanitarian crisis. But mostly their foreign policy views were taken from classical liberalism, so they they're antiwar for the most part.
In any case, when contemporary liberals call themselves progressive, their hearkening back to their intellectual predecessors.
(I discuss the underpinnings of American conservatism here if you're interested.)
EDIT: Yuval Levin has just written a book which argues that the positive rights vision of liberalism originated not with Dewey in the early twentieth century as I suggested, but rather in the late 1700's with Thomas Paine, one of America's founding fathers. Levin argues that the modern American left-right divide originated in the debate in America and England over the French Revolution. Paine was an enthusiastic supporter of the French Revolution, because he thought that government is an institution deliberately created by people to sageguard their individual liberty, and so it could be overthrown whenever the people thought it wasn't doing a good job. On the other hand, Edmund Burke, a predecessor to modern conservatism, saw government as an institution that grew organically as a tradition, and that we should be suspicious of making changes to it just because individuals in the current generation happen to object to it. In any case, Paine's philosophy of individualism led him to believe that an individual can only realize his full potential if he's not constrained by material want, so he believed in the government providing basic necessities to everyone. Thus, Levin argues, the early twentieth century Progressive movement was merely reinventing the wheel.