I'm also gonna add to Quatrarius and rob05c if that's okay.

Why do only the two parties matter? Solely because they're the biggest (I hear a lot about them and barely anything, even in history, about any other)?

Q made reference to first-past-the-post. Without proportional representation (which is used in, say, the UK I think), the goal becomes to just get as many votes as possible. To bring as many votes as possible, what in a different system would be seperate political parties compromise to form a larger party that will receive more votes. Further, because the system also benefits incumbents (in confusing ways, but just trust me that they exist), minor parties have difficulty effectively challenging the larger parties. And because they need not just some percentage, but a plurality of the vote, they simply cannot win on a federal level.

Another consequence of the system is the parties are actually rather ideologically close together. Sanders is considered rather radical, and he's just a Democratic Socialist (and a rather lean one at that). No one is arguing for new forms of government or huge fan economics changes. This isn't Hitler vs. Stalin, it's two neighbors arguing politics at a potluck.

Why do they hold the first primaries? Why is the order of the primaries as it is currently? Clearly, it's not alphabetical, and I don't have a map of the US to confirm that it's geographical.

Q nailed this. Those states literally say they have to be first. Iowa has a caucus, which is basically having large meetings of party memebers to discuss and decide on a nominee. New Hampshire has a primary, which is just a regular old vote to choose the party's nominee. Based on their results, states send delegates to a national convention to vote for the nominees selected in the state process. Interestingly, the Democratic Party uses "superdelegates," which are party officials and leaders, like former president Bill Clinton, who can vote at the national convention. The hopes is that they can influenece the nomination process to makes sure the nominee can work in the party's system.

It is important who holds the first primaries because candidates who lose the early ones will often drop out. So, not losing in Iowa and New Hampshire becomes more important, as indicator of the full national campaign.

What does the general population think of her?

A lot of the general populace, myself included, have the exact opposite opinion of her. I don't trust her at all, and never have, and some recent issues and scandals have not endeared her to me or many other moderates. However, as a group, we'd probably prefer her to Sanders, but I'd bet most moderates vote Republican in the general election. Unless Bloomberg runs.

Why do Republicans have so many presidential candidates? Is it because the last president was Democrat?

Q nailed this too, but I'll add that the Republican party has been under internal tension with the rise of a further-right contingent know as the Tea Party. They actually managed to make the Speaker of the House, a Republican, step down at the end of last year.

Finally, is it "president" or "President" in English?

rob05c exactly. President capitalized is the person, who holds the office of president.