A presidential primary is an election where members of the electorate can vote for a candidate for the Republican or Democrat presidential nomination. In an open primary, any registered voter can vote. These are favored more by the democratic party, whilst the republican party usually runs a closed primary, where only registered republicans can vote.
Each Candidate wins delegates (how many depends on the system, whether it is proportional or winner takes all) that nominate them at the National Convention, at the end of the primary election cycle. The candidates with the most delegates by Super Tuesday make them their respective party's nominees.
The main advantage to a primary election is it's a democratic process. It enables the people of America to decide on a candidate, and makes the final two the best choice for President. The review of candidates by the media and by the people also gives each candidate a mandate into the presidential election. It also shows the initial support for a candidate without conducting poll after poll. This early support can translate into support at election day and show a candidate to be popular with all the people. The benefits of primaries are that the candidates have already taken part in a democratic process and their staffs endurance, as well as their own, is tested.
A phenomena has developed and been dubbed the "invisible primary." This is a contest fought out entirely for and by the media, involving very few people except party activists and chat show hosts, making it elitist and undemocratic. The invisible primary for the presidential election cycle began on January 25th of 2008. The very afternoon Barack Obama was sworn in. His first State of the Union, Address some days later, was replied to by a very political figure. It was a young, black republican senator. Commentators argued that this meant the invisible primary had already begun, and the senator in question did neither confirm nor deny that he would be running for President. The implications of the invisible primary are that a President is constantly campaigning, and their prospective opponents are filling the airways with speeches, chat show interviews or book signings.
Overall, primaries involve the electorate early on in the presidential cycle, they increase democratic involvement and give a clear nomination by the National convention. However -- and it's a big however -- they make the election cycle incredibly long, especially given the invisible primary that takes place before Iowa and New Hampshire cast a ballot and is fought in the media. This in turn makes elections very expensive, plus the voters get bored and don't turn out come November and certain states are given undue influence in the shuffle leading up to the conventions.
It seems to me that politicians should be limited to a period of 12 months leading up to an election, in which to spout rhetoric, glad-hand and cajole voters. That should be plenty of time for skeletons to come rattling out of closets, accusations and recriminations to be made back and forth across the aisle and for voters to get a true picture of who each candidate is before deciding their vote. Granted, it might put some political talk shows out of a job, but I, for one, could do with a little less "He said, He said" and a little more of the actual meat of the candidates beliefs, motivations and plans for the future.