The general difference between these two parties is that the Republican Party is a conservative party and the Democratic Party is a liberal party. Of course, there are variations within each party, but this is an accurate generalization.
Republicans generally believe in having the government stay out of the economic sphere. They believe that economies function best when markets are left to function on their own. On the other hand, they believe in greater government intervention in people's personal lives. They are in favor of things like government efforts to promote abstinence or to encourage religion.
Democrats are generally the opposite of this. They tend to believe in leaving people alone in their personal lives, but they believe in government regulation of the economy.