How do the US elect their President?

What you're seeing now (with all the placard waving) is the NOMINATION process.

Organized political parties have a nomination process to determine who they will support during the election. The national party organizations determine the rules for the 'convention' where state level delegates vote to determine the nominee. At the state level, the state legislatures determine the rules for the state delegate selections. (Generally either a primary - secret ballot - or a caucus - public ballot meetings ... although a few states have a combination of both.)

Once the delegates are selected, there is a party convention where the final selection of the party nominee occurs.

The nominees then face off in the election. Every state votes on the same day (first Tues in Nov). The ballot shows the name of the person you want as president, but in reality you are voting for 'electors'. (State legislatures, again, decide how they will divvy up the electors based on the popular vote. The majority of states have a winner-take-all assignment of electors, but other rules have been proposed in a couple of states.)

Then in December, electors then cast their votes, and the results of the 'electoral college' votes determine who is actually elected president.

Yeah. It IS that messed up.

A bunch of autismal candidates go across the country telling the public bullshit, which ever bullshit sounds the best has the majority of the vote, once to votes are sent into Washington District of Corruption otherwise known as "D.C." it is processed by paper cutters and the corporations and the military decide who should be the president. And thats how that monkey Bush got into office twice.

It varies, sometimes the vote is overwhelming and the people get who they voted for. Sometimes the voting is close, then the Supreme Court picks a winner based on which party appointed them to the bench.

Wikipedia -> Electoral college.