The Constitution allows impeachment of the president for abuse of the authority of his office, and also for obtaining his office by corrupt means.
Cass Sunstein provided the history behind the impeachment clause in an opinion piece last year:
[The impeachment clause] says that a president may be removed only for “Treason, Bribery, or other high Crimes and Misdemeanors.” The reference to treason and bribery, together with the word “other,” suggests that the president must have violated the law — and that the violation must be quite egregious (“high Crimes”).
But the debates at the Constitutional Convention offer a radically different perspective. An early draft of the founding document said that the president could be impeached for “malpractice, or neglect of duty.” That draft provoked an extended debate, featuring three distinct positions.
An extreme view, attracting little support, was close to [President Gerald] Ford’s: The legislature should be able to remove the president at its pleasure. An equally extreme view, obtaining considerable support, was that the president should not be impeachable at all. The third position, which carried the day, was that the president should be impeachable, but only for a narrow category of egregious abuses.
Promoting that compromise in 1787, George Mason argued that the Constitution must allow a response if the president obtained office by corrupting his electors. That argument led other delegates to agree that impeachment might be permitted in situations of “corruption & some few other offences.”
James Madison concurred, pointing to cases in which a president “might betray his trust to foreign powers.” Gouverneur Morris added that the president “may be bribed by a greater interest to betray his trust; and no one would say that we ought to expose ourselves to the danger of seeing the first Magistrate in foreign pay without being able to guard against it by displacing him.”