If the name anyone who wanted to run for office appeared on every ballot, then election ballots would be unmanageably long.
For high-profile elections, there is almost always a democrat and republican candidate, and there is normally a number of other parties' candidates.
Often, lesser-known parties will only appear on some ballots, but all of them.
How is it decided which names appear on a particular ballot?
The actual procedure varies from state to state.
But in general a candidate needs to get a certain number of unique signatures on a petition supporting their candidacy. Most states allow parties that received a certain percentage of votes in the previous election to nominate a candidate in the next election. Parties can do this how ever they choose. The GOP and the DNC do this through a primary election process.
In theory anyone could attempt to run by collecting enough signatures to get on the primary ballot but the reality of election politics today is that only those people who are connected to the party already and have the ability to raise money for themselves and the party have a real chance of success of being nominated to a high profile position.
Many of the minor parties maintain a list of people willing to sign a petition for their candidate. Usually these parties select a candidate from their ranks though occasionally a celebrity or someone who has been elected before as a candidate of a major party but has become disgruntled with their party will turn up running for one of the minor parties.
To get involved in politics and eventually rise to a high profile position most people start out either in a local or state board. State boards are often filled with interested business people who have spent some time working with and often contributing to one of the major parties. Local boards, especially ones that are elected are where the grass roots candidate can get their start. But the people who grow from local boards get involved with helping a party out as they through fund raisers and other events.
@Chad is correct that particular procedures will be determined on a state-to-state basis, and because of this, there are no hard-and-fast generalizations to make. The way Texas does it (where I'm knowledgable) is thusly:
For major-party candidates (Democrat or Republican), prospective candidates file with the county party organization in the county of their residences, and then the parties will conduct a primary whose winners will be placed on the general election ballot.
Minor-parties can do it basically the same way. However, there are some gotchas for the guy who wants to put a sign on his garage and call himself a political party. In Texas, minor parties have to meet certain requirements to be considered a political party:
Other details can be found in the candidate guide I linked. Minor parties nominate general election candidates in the same way that major parties do. If the party's nominee for gubernatorial candidate received more than 1%, then the party is entitled to place its candidates for state and county offices on the general election ballot. If the nominee received less than 20% of the votes in the last election, they may may choose between holding a primary election or nominating by convention. If the party's candidate received more than 20% in the last gubernatorial election then they are required to hold a primary.
Independent candidates may nominate themselves for state or county offices by submitting the correct application along with a petition with a number of signatures determined by the Elections Code for the different offices. A handy table of numbers of required signatures is published at the Sec. of State site.
But Wait! There's More!
Elections for municipal or other local offices (city councils, governing boards of schools, hospitals, etc) are nonpartisan by law, with the exception that home-rule cities may legislate partisan elections for themselves. These positions do not require petitions generally (with similar exceptions), and only require an application to be filed with the appropriate electoral authority (i.e., with the governing body of which the position sought is a part).
