I think (election) campaign is the phrase you are looking for. After a candidate is nominated he runs a campaign to try to convince voters to vote for him in the upcoming election.
primary
They are "elected".
primary election
A preliminary election where delegates or nominees are chosen is often referred to as a primary election. In this process, voters select their preferred candidates from within a political party to compete in the general election. Primaries can be either open, allowing any registered voter to participate, or closed, restricting participation to registered party members. The outcomes of these elections play a crucial role in determining the candidates who will represent each party in the general election.
Vice presidential candidates are generally chosen in order to garner extra votes in the general election. The votes will usually come from a specific area of the nation like the South or the West.
The president is elected by direct election every 5 years. If no candidate receives a majority of the votes, a run-off election is held between the top 2 candidates.
An election in which delegates select and nominate candidates is called a caucus.
No, he was one of the candidates for the presidency during the 2012 election, and he won. There is no election in 2013.
A federal election begins with the announcement of the election date, typically set by the government or electoral authority. This is followed by the issuance of official guidelines for candidates, including eligibility and nomination processes. Political parties and independent candidates then prepare to campaign, and voters are informed about registration and voting procedures. The election culminates in a designated voting period where citizens cast their ballots.
In the Jeffersonian democracy presidential candidates would be chosen by caucuses that were held by political leaders. In a Jacksonian democracy the candidates would be chosen by conventions.
In a primary election, voters choose candidates who will represent their political party in the general election.
Sure, and in fact they often do. They wouldn't do so near the general election, because by then most of the time each party has its candidates already chosen, and there's not really much point in having the guy you've already chosen debate a bunch of guys you passed over. Before the primaries, though, it does make sense to have the candidates for each party debate each other.