What these parties have in common is a belief in the basics of the American system. Both believe in a capitalist economy and a democratic goverment with free electionss and civil rights and civil liberties for all.
Outside of those basic ideas, there are a lot of other ideas. Here are some differences, with the Republican idea first in each case.
- More laws about morality vs. letting people choose for themselves. Republicans want the government to do things like prohibiting gay marriage while Democrats don't.
- Fewer regulations on the economy vs. more. Democrats want lots of rules to prevent big companies from hurting our society. Republicans think the rules hurt us more.
- More military vs. less. In general, Republicans (though not the Ron Paul branch) want to use our military force to get what is good for us. Democrats want us to use diplomacy more.
- Less government help for people vs. more. Republicans believe the government should not really provide welfare or much in the way of unemployment benefits because they give people the incentive to be lazy.
There are many other differences, but these provide a basic view of the philosophical differences between the Republicans, who are conservative, and the Democrats, who are liberal.