Primaries are the system in which political parties nominate a candidate for a given office. The office in question varies from local and state offices to federal and presidential. Not all offices have a nominee from every party, or even from the two major parties (think people running unopposed for local positions). Primaries do not happen for non-partisan offices such as judges (when they are elected - often they are appointed). Primaries come in several forms.
The best known caucus is the Iowa presidential caucus, though there are caucuses in other states for the presidential election as well as for other political offices. In Iowa, one finds a complex, time-consuming process in which supporters of the various candidates gather to select the party nominee. A precinct is given a numerical score based on its turnout in the last presidential and gubernatorial election. Based on that score, delegates are alloted to a particular precinct for the county convention, which is irrelevant because it is held after other states have decided the primary election. The candidate with the most delegates takes the Iowa presidential nomination. Inside each precinct, a presidential candidate must have a certain percentage of the vote in order to be viable for a delegate (15% in most precincts, since a majority of precincts allot more than 3 delegates).
Several states use the caucus systems and the process varies by state, with some states allowing caucus goers to attend anytime during the day and state their preference.
As noted above, a caucus is very time consuming, much moreso than an election. In Iowa, for example, the process can take up to three or four hours. This reduces the rate of participation in the primary, as potential caucusgoers may have to work or tend to children during caucus night. Furthermore, many potential caucusgoers simply find the process to be very boring. This also serves as a deterrent to participation.
Simpler process in which voters go to the polls and cast an anonymous vote for the candidate of their choice.