The major undemocratic aspects of our system that were included by the Framers of the Constitution are the nature of the judicial branch of the government , the way in which the president is elected,  (the Framers also had an undemocratic way of electing senators, but we did away with that by changing the Constitution in the 1910s) and the makeup of the Senate.

In our system, the president is not elected by direct popular vote.  Instead, he (or someday she) is elected by the Electoral College.  This used to be more undemocratic than it is now.  In the past, the people did not get to elect the Electoral College.  Now we do.  The Electoral College is still somewhat antidemocratic because (as was the case in the 2000 election) a person can win the popular vote and lose the election.  The judicial system is even more undemocratic since people do not elect the federal judges in any way and yet those judges can make decisions that dramatically affect our society.  Finally, the Senate is undemocratic because states with very few people, like Wyoming, have as many senators as states with huge populations, like California.

These provisions were put into the Constitution because the Framers did not completely trust the idea of democracy.  Democracy had never been tried before and seemed risky.  The Framers worried that the people would act rashly, without thinking.  Therefore, they wanted to have some distance between the people and the government so the government would be able to act more intelligently than the masses of more uninformed people would.