A direct democracy is where people decide and vote on policy initiatives directly
A representative democracy is where a group of people vote for an elected official to represent them and make decisions on their behalf.
The United States is a representative democracy, since we elect different representatives at the city, county, state and federal level. A direct democracy does not have elected officials, because citizens vote on their own behalf.
Ancient Athens is the best example of a direct democracy. Although they excluded women, slaves and foreigners from voting, everyone else was required to vote on all major issues.