voting laws vary in detail from voting district to voting district. They are governed by state statutes and guidelines and generally operated by volunteers. Hence the complaining about the "idiots" or one place versus the presumably "enlightened" of another ... but all that aside

Primary elections are conducted for the purpose of political parties (basically Republican and Democrat in the US) to select their single candidate for a given office - one who will run against the single candidate of another party - or in some cases parties. It is not a requirement that parties have primaries. It is simply the process by which most parties select thei candidate for a given office.

In many states voters can vote in primary elections for EITHER party - but not both. In some states you must be a registered party member to vote in the primary. In some states voting in a primary leaves a registered member of that party until you vote in another primary for another party.

Again - primaries are used to select the candidate that a political party will run for a given office. Then elections are held in which the list of final candidates is offered on a ballot - one candidate for each party, typically no more then 2 parties runs candidates - but nearly always there are more for president, and some regions have greater influence from third or fourth parties. Write in voting is also allowed - so the candidate of your choice does not have to appear on the ballot.