Difference between direct democracy and indirect democracy

An indirect democracy is otherwise known as a "republic" or a "representative democracy." Many countries that are well-known for their democracy, such as the United States, are actually republics and not true direct democracies.
A direct democracy is exactly how it sounds: the people cast votes, and the person with the most votes at the end of the election wins. If my ballot is counted last, and my vote determines if person A or person B wins the election, you can say that my vote "directly matters." This might seem obvious, but when it comes to republics, not all votes directly "matter."
In a republic, people vote for representatives. These are often governors, senators, representatives for the house, and other legislators. In these district elections, the votes of people "directly matter" because the candidate with the most votes wins. However, in political practices one tier above that, the common man does not get to directly vote.
Legislators and senators and the like are tasked with voting for their district. That means that your county has chosen representatives who have beliefs that match with most of yours. Thus, those representative people will vote in congress or court or wherever how you probably would have voted. This simplifies the process and allows for educated legal specialists (senators) to make educated legal decisions instead of leaving it up to you and me.
This also takes place within voting for certain offices, like the presidency. In America, there is an electoral college. That group of educated voters sits in a little office and takes a look at the ballots cast by the common people for the candidates for presidency. The electoral college people then take their votes (the ones that actually count) and cast them probably based off of the votes of their represented districts. So if everyone in Washington votes for the Democratic candidate, the electoral college for that state will cast their (actually important) votes for the Democratic candidate.
Now that is not to say that the common man's vote does not matter; if the common people did not cast their votes for office, then the electoral college would not know who the public wanted them to vote for. It's like telling your mom that you want a green scarf, but she ultimately has the decision and buying power. She might buy you the green one, but she also might see that the scarf is made of a scratchy fabric that might itch you, so she picks an orange one instead. She takes your input into account. So too does the electoral college.
So, a direct democracy allows everyone to participate in the "most votes" contest. An indirect democracy allows everyone to participate in the "most votes" contest to choose educated politicians to make the hard decisions for us.