A candidate runs for nomination of their party. These elections are called primary elections. A primary election is an election in which voters in a jurisdiction select candidates for a subsequent election (nominating primary).
Read about Primary Elections Here:
http://en.wikipedia.org/wiki/Primary_election
You can run as an Independant candidate, but you won't be President.
The nominations of each party and any independents run against each other in the General Election.
A President can serve only 2 terms (8 years) and then can not run again.