Wormholes in reality do not resemble their whirlpool counterparts
An Einstein-Rosen bridge, colloquially known as a wormhole, is a distortion of space-time that in theory may allow near-instantaneous transit to anywhere in the universe. It is named after physicists Albert Einstein and Nathan Rosen, who are popularly credited with developing the theory of wormholes. In the film, a wormhole presents the key to humanity's chance for survival by allowing the crews of the Lazarus Project and later the Endurance to travel light years away to another galaxy containing potentially habitable worlds.
History
NASA observed stuff via space telescopes the existence of a disturbance in space near Saturn, about 48 years before the start of the film. This gravitational anomaly was discovered to be a wormhole. The wormhole is still near Saturn when Cooper exits the Tesseract, about 137 (Earth) years after its discovery.
The wormhole puts potentially habitable worlds within humanity's reach for colonization. Thirty-eight years after its discovery, NASA launched the Lazarus missions, and ten years later the Endurance, to ensure survival of the human race on one of these worlds.
As wormholes are not naturally occurring phenomena, NASA theorized that the wormhole was placed there by unknown beings looking out for Earth's population, giving them the opportunity to find a new world. Cooper theorizes that the wormhole was placed there by future humans in order to ensure the survival of their species in the past.
As the Endurance approach the wormhole, Cooper expresses surprise at the wormhole's spherical appearance, noting that all of the text-book drawings he had seen of wormholes presented it as a circular vortex. Romilly explains that this is due to the universe being represented in two dimensions, and thus a sphere would be represented as a circle. Romilly further explains that wormholes function by "folding" space in half and allowing an object to travel from one side of the wormhole to the other instantaneously, representing this concept by bending a sheet of paper in half and pushing a pencil (representing the Endurance) through the paper.
Once they have reached the wormhole, Cooper flies the Endurance around the sphere and gradually edges the ship towards the wormhole's surface until it is pulled inside. Once inside, the Endurance is propelled through the wormhole's passage, which appears as multidimensional passage composed of cosmic matter that is in a constant state of flux. Cooper discovers that he has lost control of the Endurance, and Doyle explains that this is due to the fact that the ship has passed through the "bulk," or space beyond human's three dimensions. As the crew of the Endurance rockets through the wormhole, a visual distortion in space appears inside of the ship's cockpit, which Amelia Brand believes to be the bulk beings making contact with the crew. Despite Romilly's protests, Amelia reaches out to touch the distortion, which causes her hand to begin warping. Before they can make any further determination as to the nature or origin of the distortion in space, the ship exits the wormhole, although Amelia states she believes she has just engaged the the "first handshake" with extraterrestrial life (although the distortion is later revealed to be Cooper reaching out to Amelia as he travels through the black hole after exiting the Tesseract).
Later, after Cooper allows himself to fall into Gargantua in order to gather the data necessary for Murphy to solve Dr. Brand's equation, he enters the Tesseract, where he realizes that unknown beings who created the wormhole are in fact humans from the distant future. These humans had evolved to the point that they had mastered the laws of physics and existed within five dimensions, possessing the ability to create wormholes and placing one near Saturn in order to give humanity a chance to save itself from their dying world.
2008 Script
The Bulk, the region inside a wormhole.
The opening sequence of the script features the collision of a neutron star and a black hole which creates an energy burst that ends up detected by LIGO receivers, this is the first discovery of the wormhole.
Speculation
It is possible that the wormhole became a permanent 'moon/satellite' of Saturn.