The main thing to remember about the American political system is that it is not a parlimentary system. That is to say the chief executive (President and state Governors) are elected separately from the legislatures (Congress and state legislatures). As a result, party discipline is much less important in the U.S.
Because of this the President (or a Governor) is frequently of a different political party than one or both houses of Congress (or the state Legislature). In order to pass legislation, it must be acceptable to at least a majority of the legislators of both houses as well as the President (or Governor). On most legislation, there are at least a few legislators that vote the opposite way from the bulk of their party.
One thing that we don’t have here that you do have in most parlimentary systems is a “no confidence” vote with the resultant early elections. Executives and legislators are elected for fixed terms, and early elections cannot be called. Because of this, politicians know when the elections will be held and can begin campaigning for particular offices long before the elections (years before in some cases). Because there will frequently be multiple candidates from each party interested in running, there can be a party primary election. In that election, the voters (usually only those registered to the relevant party) will vote for the candidate they prefer to be their party’s nominee.
For most offices (Congress, Governors, state legislatures), the winners of the primaries challenge each other in the general election (held the first Tuesday in October), and the candidate with the largest number of votes wins.
For President, it is a little different. The Presidential candidates for the main political parties are selected at a convention held in the summer, with delegates from each state voting for their candidates. Although most delegates are “pledged” to a candidate (usually the candidate that won their primary, though there some complexities that I won’t get into here), if the pledged candidate does not get a majority of delegates on the first ballot, they can change their votes on subsequent ballots. In recent years, there has not been a party convention in which the winning candidate was not known well before the convention because he had the overwhelming majority of delegates.
Once the Presidential candidate is selected, he gets to select his candidate for Vice President, known as his running mate. In November, you will vote for a President/Vice President ticket, and cannot separately vote for the individuals. (Actually, you are voting for electors pledged to the President/VP ticket, as mentioned in Lsura’s post). Whichever set of candidates has a majority of votes in each states gets all of the electors in that state.
After the election, the electors for each state meet, and the electoral vote is transmitted to Congress. (Note that the “electoral college”, never meets as a body, and exists only for the purpose of voting for President once every four years.)
If a set of candidates gets a majority of electoral votes, they become President and Vice President. If no candidate gets a majority of electoral votes, the House of Representatives gets to select the President from among the top three electoral vote-getters.
Or, to summarize a confusing process, in November we will be voting separately for the President/Vice President ticket (or actually electors pledged for the ticket), our local Congressperson (with Congress member being elected to a two year term), and in two-thirds of the states, for one Senator (with each Senator being elected for a six year term, with one-third of the Senators up every two years).