Each candidate is elected by the primary process. Each of the 50 states, and some territories, selects a date that is acceptable to the respective national committee for its Republican and Democratic primary or caucus (or both, although for every state except Texas, the primary is irrelevant). Generally, the committees allow some states to go before a certain date (Iowa and New Hampshire have always been first) and every other state has to go on that date or later (this year it was February 5; many states tend to go on this date, which is always a Tuesday, so it's called Super Tuesday). Each state is awarded a set of delegates, based mostly on population, but sometimes there are other factors (North Carolina got 20 extra delegates for holding its primary very late).
In a primary, voters go to voting booths and vote like in a normal election. The tricky part is how delegates are allocated based on the vote, which is determined by each state. California, for instance, gives a chunk of delegates to the statewide winner, plus 3 delegates for winning each county. In the Republicans' system, some states are winner-take-all, meaning, if you win the state, you get every delegate (John McCain got every delegate from New York due to this rule, which helped put him on top). Democrats don't do this, though - which largely explains why neither Obama nor Clinton was ever able to put the other away.
Then there are caucuses... those are variable, so I'll just link to Wikipedia's explanation of the
Iowa caucuses, the most well-known.
There are also superdelegates - generally, these are anyone the national committees see fit to give a vote at the convention, and they can vote at the convention for whoever they want. In reality, these tend to be elected officials and well-known party names (Bill Clinton and Al Gore are superdelegates).
In the end, all of these votes are tallied up at the convention and, whoever gets a majority (there are revotes until someone gets a majority; pluralities are not enough) is the party's nominee.
