The simplest way to put this is the way my government textbook put it:
Democrats are socially liberal, and are to the left on economic issues.
Republicans are socially conservative, and are to the right on economic issues.
Basically, Democrats like freedom when it comes to social issues, and Republicans like freedom when it comes to economics. Of course, there a plenty of exceptions to these guidelines, and neoconservatives completely destroy that definition of Republicans.
If you want a real understanding of American politics, I think it's best to think in terms of who supports the parties. The two major parties don't really have ideologies-- they've historically pulled 180s on their platforms in response to demographic changes. Republicans and Democrats are defined by the interest groups who bribe them. Republicans are bribed by corporate interests and Protestant evangelicals, and Democrats are bribed by labor unions and persecuted social minorities (blacks, gays, etc.). The reason these two parties have survived so long is because they are so malleable; they're meaningless words used to describe the group in power and the opposition, whoever that may be.