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 incredible ability to open any lock that was made in America. It may have been part of the reason that Sheriff Blubs and Deputy Durland were looking for Trembley.
History
The short-lived eighth-and-a-half president Quentin Trembley was given the key when he became president. However, his presidential term was viewed as an embarrassing accident due to his rampant silliness, and he was thus kicked out of office. 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 detain them, in order to keep the nation's greatest embarrassment a secret. They took the president and the twins by train to Washington, D.C. However, on the way to the city, Mabel accidentally helped the former president escape from his hibernation. Realizing they were in a crate, he attempted to use the key to escape, to no avail.
After they escaped, Trembley gave the key to Dipper along with the -$12 dollar bill, telling him to go unlock America's secrets. Dipper was able to use the key to unlock Grunkle Stan from the stockades in which he had been placed.
Use
Dipper gets the President's Key from Trembley
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.