One of the big differences between the US voting system and most (if not all?) of the European systems I am familiar with is that in the States, we literally vote for the person, rather than the party the person represents.
Campaign ads for candidates in any race, from president to a local city representative, almost never mention the party the person represents. The ads focus on that individual’s qualities, beliefs, and/or plans (or they focus on the negative qualities, beliefs, and/or plans of the opponent), but there is rarely even a footnote to tell viewers what party the person represents.
You can usually get this information from the local newspaper, and it is generally listed on the ballot when you actually vote, but there are even some positions that are completely party-free, such as local judges.
As has been explained already, the populace actually chooses the final candidate for each party in the final elections, rather than having the choice left to the party itself. ANYONE who has the desire and financial resources can run for any office, for any party, and the person with the most votes in the end wins, regardless of any background experience or qualifications. This is why we sometimes end up with rich celebrities in positions of governmental power, including actors, soldiers, and astronauts. (Note: I am not claiming that none of these people are qualified, but it is true that the States have a higher proportion of rich celebrities in political offices than most other democractic countries.)
Party affiliation is also somewhat of a joke here, since both of the major parties are pretty centralist. The Republicans tend to be slightly right of center, and the Democrats tend to be slightly left of center, but the differences often aren’t significant except in one or two areas, such as abortion. (Republicans are more likely to be anti-abortion, while Democrats are more likely to be pro-choice, but even this isn’t a hard and fast rule.)
Even after being elected, the person holding office makes decisions about policy, rather than having the policy dictated by the party they represent. It’s quite possible to have a Republican who is publicly pro-choice, or a Democrat who is publicly anti-abortion, even though the party lines tend in the opposite direction. It is also possible for an official to switch parties while still in office. This is rare, but not unheard of, and while the individual may not be re-elected the next time around, it does not mean automatic loss of position, as it would in Europe.
In the States, many voters are affiliated with a specific party based on their registration choices and/or their own political leanings, but only a relatively small percentage of the population actually joins one of the parties, to pay dues. Even when an American declares that s/he is a “card-carrying” Democrat or Republican, that usually means only that s/he has checked that particular box on her/his voter registration form, and the party affiliation is noted on the card they issue to tell you where you have to go to vote. No money changes hands for this “privilege” in most cases, and the individual voter is free to vote for any candidate in any final election. (The right to vote in primary elections has already been discussed.)
Our Federal tax forms, though, do include the option of contributing $2 per taxpayer to the Presidential Fund, which provides campaign funds to any candidate running for the office of President, regardless of party. I usually check the box to allow this, primarily to give non-party candidates the chance to run for office.
When a candiate “represents” a party, it simply means that they can use financial resources for that party, including especially donor contributions to that party. The Democratic party, for example, will pay for a certain number of campaign ads for candidates who are affiliated with the Democratic party. The Democratic party could refuse to provide funding to candidates who “claim” to be Democrats, but whose policies are contrary to those that are generally accepted by the Democratic party, but if said candidate has enough money and/or individual donations to run the campaign without the financial support of the Democratic party (which is admittedly rare), there is really nothing to prevent that person from running as a Democrat on the ballot. In most situations like this, though, the candidate will run as an Independent, to avoid conflicts.