Presidential preliminaries - explanation

At the moment, we are in the Preliminary/Primary phase of voting. everyone from the Democrats or the Republicans who wants to run for president puts their hat in the ring, and then they travel the country raising money (lots required) and holding rallies, and trying to impress. There are about 15 Republican candidates, I'm not sure on the democrats. You can only vote in the Primary round of voting if you are a registered member of one of the two main parties. Then, the two winners of that round go up against each other, as well as any independent candidates that may have declared, and everyone votes again.
This is roughly how I understand it works.
Australian way - the party (or party membership and the party, if you're labour) deciding who is the leader of the party, and then people voting for their party, and the party with the most seats winning government, and the leader of that party being the Prime Minister/leader of the country.
American way - way for party members across the US to vote for who is going to be the best (or who has spent the most money, and therefore looks like the best) person of the two main parties to progress through to to the final round of presidential vote. The presidential vote is separate to the voting for representation in the congress. The president does not have to win a seat in congress. They are one of the three branches of federal government, the other two being the people in congress and the judiciary.