The President of the United States is eligible for two four-year terms.
After serving one term, the President may choose to run again. Incumbent Presidents are often elected, but not always.
Upon completion of the second term or a loss after the first term, many former President head into the private sector to make money giving speeches and peddling influence. They may open a Presidential Library where the artifacts from their Presidency will be stored for historians to study. They may write books. They may start a charity.
There is no one right way to be a former President.