For a sitting president, the only "automatic" disqualification is death.
Everything else requires action by the President or other parties.
Section 1 of the 25th Amendment to the US Constitution has the relevant text:
In case of the removal of the President from office or of his death or resignation, the Vice President shall become President.
Taking those in reverse order:
- Resignation clearly is a voluntary act by the president.
- Death is obvious, I assume a doctor would make this official.
- Removal is slightly more complicated, let's examine this one in more detail.
The obvious mechanism for removal of a sitting US President is impeachment, a process covered by other excellent answers here. In short, the US House of Representatives votes on "Articles of Impeachment" against the president (by simple majority), and the US Senate holds a "trial" to examine the evidence and render a verdict. A 2/3rds majority vote is required to remove the President.
The only other proposed mechanism for removal is via Section 4 of the 25th Amendment. This section has never been invoked, but allows the Vice President along with other administration officials to send a written declaration to Congress declaring the President "unable to discharge the powers and duties of his office". Even this, though, seems to imply a temporary removal, not a permanent one; the Vice President merely becomes acting President during this time.
EDIT: Even if it was discovered, after taking office, that the President didn't actually satisfy some of the constitutional criteria for the office (turns out they were actually younger than 35 when elected, or they weren't actually a natural born citizen, for example), there is no automatic mechanism for removal or disqualification, they would have to be impeached to be removed.