A presidency is the executive office of a country, state, company, or other large organization. It's also the way to refer to the whole length of time a president is president: "Washington's presidency ended after two terms."
When a candidate is elected to the presidency, that person becomes president, the head of a nation, organization, college, or company. The presidency of the United States has potential to change every four years, when the country holds elections. You can also use presidency to talk about a particular time period, defined by who was president them: "His presidency2020欧洲杯时间 was a period marked by recession and job loss."