The President's Key is an item that was presumably handed down to each president of the United States. The key was officially last held by Quentin Trembley until he was kicked out of office and disappeared with it. The key has the ability to open any lock in America.
The short-lived eighth-and-a-half president Quentin Trembley was given the key when he became the president of the United States. However, his presidential term was viewed as an embarrassment due to his rampant silliness, and he was thus kicked out of office, and all knowledge of his term covered up. He was able to escape to a small valley in Oregon, where he would found the town of Gravity Falls. The key was lost with him.
Years later, his body, preserved in peanut brittle, was uncovered by Dipper and Mabel Pines. Sheriff Blubs and Deputy Durland, however, had been ordered to find the body, and ordered to detain them, in order to keep one of the nation's greatest embarrassment a secret. They took the president and the twins to Washington, D.C. by train. However, on the way to the capital, Mabel accidentally helped the former president escape from his hibernation. Realizing they were in a crate, he attempted to use the key to escape by smashing it into the wall, but to no avail.
After they escaped, Trembley gave the key to Dipper along with a Negative Twelve Dollar Bill, telling him to go unlock Gravity Falls' secrets. Dipper was later able to use the key to unlock Stan Pines from the stockades in which he had been placed.
In "The Last Mabelcorn," while he was connected to the Project Mentem device created by Ford Pines, the President's Key was among Dipper's numerous thoughts displayed on the monitors. Humorously, he briefly considered the possibility of robbing a bank with the key.
The President's Key functions similarly to a skeleton key, in that it can be used to open any lock, regardless of shape or size. However, from Trembley's wording, it can be inferred that it is mildly limited in that it can only open locks made in America. Dipper was able to successfully utilize the key to break Grunkle Stan out of the stockades that he was locked in by Deputy Durland and Sheriff Blubs for being insubordinate.