The difference between direct and indirect democracy has to do with how directly the people get to participate in the democracy. As the names imply, the people have much more direct participation in a direct democracy. They participate so directly in the democracy that the system is essentially unworkable and there are no countries that have direct democracy in the world today.
In a direct democracy, the people actually make laws for themselves instead of electing representatives who will do it for them. They have to propose laws, consider them, and finally vote on them. This is different, of course, from our own system in which we elect representatives to make our laws for us. We do that at every level from the town level to the national level. Our system is an indirect democracy.
Direct democracy is generally understood to be impossible in the modern world. It would only be possible in a polity that had a very small number of people who were allowed to participate. There are (as the link below shows) those who believe that technology now gives us the ability to have a direct democracy, but this has yet to be proven.
The difference between them, then, is that people elect representatives to make laws for them in an indirect democracy whereas people make their own laws in a direct democracy.