The eligibility requirements to run for the office of President are:
Once a candidate enters the election, what criteria determines if the candidate is in fact electable enough to appear on the election ballot? Does this vary by state, or is there a criteria that all states follow?
These are generically called nomination rules; the US calls them ballot access rules. They're entirely up to the states and can pretty much be anything, although unfair ballot access rules can be ruled unconstitutional (this happened in Tennessee earlier this year).
In fact, the entire process of choosing electors is left up to the states, they just need to somehow choose the right number of electors and then have them vote. From the US Constitution, Article II, Section 1 (emphasis mine):
Each State shall appoint, in such Manner as the Legislature thereof may direct, a Number of Electors, equal to the whole Number of Senators and Representatives to which the State may be entitled in the Congress
Currently all states except Maine and Nebraska use the same system, where all electors are supposed to vote for the plurality winner in that state's count, but states are free to choose electors however they want, so they don't even need to have a ballot in the traditional sense; as far as I can tell it would be legal for a state to decide the Governor is going to choose that state's electors and they're not going to bother with a popular vote anymore. This is what makes things like the National Popular Vote possible
In the originally-envisioned system, Presidential electors are appointed by the states; the states decide how those electors are chosen. Accordingly, states could hold an election, but in most the state legislature chose the electors. Since the Civil War, all states have chosen electors according to popular vote, but ballot access rules (for the general election) are still determined by the states.
At present, Presidential primaries are theoretically run by the parties, who can make their own ballot-access rules. (But they may be subject to restrictions under state law.) Effectively, major-party candidates must follow their party's procedure to gain primary ballot access; third-party or independent candidates (or major-party candidates that lost their primary but want to continue their campaign) must follow the state procedure.
Typical requirements include submitting some number of registered voters' signatures and paying a filing fee; alternatively a candidate nominated by a party that received some percentage of the vote in the preceding election often automatically gets ballot access. The intent is to ensure that only serious candidates get on the ballot. Allowing large numbers of frivolous candidates would confuse voters, and increase the chance that the first-past-the-post system elects an undesired candidate (by splitting the vote between similar candidates). (Consider comedian Stephen Colbert's candidacy in 2008; he was denied primary ballot access because his candidacy was deemed to be a joke.)
A candidate/party that lacks ballot access can still be elected by write-in votes, but many states require write-in candidates to submit forms first.
A bizarre case where the ballot-access rules went wrong was a Tennessee State Senate election in 1998, in which candidate Byron Looper removed his only opponent from the ballot by assassinating him (Tennessee law denied ballot access to deceased people).
It is worth noting that some states even provide ballot access to constitutionally ineligible candidates that meet the state requirements. For example, for the 2012 election, the Party for Socialism and Liberation nominated a 28-year-old and a foreign-born citizen for President and Vice President, but the ticket nevertheless obtained ballot access in nine states.
