The Space Stone is one of the six Infinity Stones, the remnant of a singularity that predates the universe. Housed within the Tesseract, over the course of history it was wielded by many individuals, including Johann Schmidt of HYDRA and Loki of Asgard. During the Infinity War, the Tesseract was broken by the Mad Titan Thanos, who put the Stone inside his Infinity Gauntlet.
After the universe came to be, the Cosmic Entities fashioned the remnant of six singularities into concentrated ingots which were later dubbed the Infinity Stones. Befitting its name, the Space Stone held dominion over the physical stuff of space being able to teleport its users anywhere in the universe. The earliest known people to possess the Space Stone were the Asgardians who ensconced the Stone within a crystalline containment vessel known as the Tesseract as a means of controlling its formidable power. Circa 965 A.D., whether by accident or intentionally, the Asgardians left the Tesseract on the planet Earth. In the legends of Norse mythology, the Tesseract came to be regarded as "the jewel of Odin's treasure room."
World War II
The Tesseract resurfaced in 1942 where it was procured by Johann Schmidt who sought to use its power to mass produce a line of advanced weaponry. The power he managed to extract was beyond what he imagined and gave him enough confidence to turn his back on the Third Reich entirely and make his own plans for world domination. In 1945, Schmidt managed to refine the Tesseract's power into a series of devastating bombs and a personal aircraft through which he could achieve world conquest. His plane was boarded by Captain America with the chaos of the fighting shaking the Tesseract free from its restraints. When Schmidt seized the artifact, it activated and, in disgust of his abuse of its power, banished him to Vormir, resting place of the Soul Stone, knowing he would never be able to make the sacrifice required for its power. The Tesseract then burned through the decks of the Valkyrie, eventually coming to rest in the Arctic Ocean where it was recovered by Howard Stark.
After helping to found S.H.I.E.L.D., Howard Stark studied the Tesseract for some time and eventually theorizing a new element based on its atomic structure. Owing to the technology of his time, he was unable to actually synthesize the new element instead substituting it with Palladium in his Arc Reactor prototypes. Howard left detailed notes on both the Tesseract and his new element in the hopes that his son would one day succeed where he had failed. Following Howard's passing, the Tesseract remained in S.H.I.E.L.D.'s care. By 2010, the World Security Council had sanctioned Project P.E.G.A.S.U.S., an attempt to harness the Tesseract's power into advanced weaponry. Following Thor's arrival on Earth, S.H.I.E.L.D. hired Doctor Erik Selvig to study the alien artifact and accelerate their progress.
Following his defeat by Thor and banishment from Asgard, Loki found himself in the territory of Thanos who agreed to supply Loki for a Chitauri army with which to conquer Earth in exchange for the Tesseract. Via remote manipulation of its energies, Loki teleported himself to Earth and brainwashed several S.H.I.E.L.D. agents and personnel, including Selvig, to safely contain the Tesseract's energies to open a stable wormhole for the Chitauri to invade through. Though the Chitauri made planetfall, they were defeated by the newly-formed Avengers with Thor taking the Tesseract and Loki back to Asgard. Thor then entrusted the artifact to Heimdall who used it to repair the Bifrost Bridge before storing it back in Asgard's vault.
As the recently freed Hela inspected the treasures in the vault, she admitted the Tesseract was "not bad", but dismissed it in favour of the Eternal Flame. When Loki sought to enact Ragnarök, he passed by the Tesseract and took it with him as Asgard was destroyed. Detecting the Tesseract aboard the Statesman, Thanos had the Sanctuary II intercept the refugee ship.
In Thanos' Possession
After slaughtering most of the remaining Asgardians, Thanos tortured Thor with the Power Stone to force Loki into surrendering the Tesseract. Though Loki attempted to use the Tesseract as a distraction long enough for Hulk to defeat Thanos, the Titan managed to best the gamma-beast and claimed the Space Stone, shattering the Tesseract in the process. After placing it in the Infinity Gauntlet, Thanos used its power to halt one of Loki's daggers before teleporting himself and the Black Order off the Statesman after using the Power Stone to destroy it.
Though continuing to use the Sanctuary II as a home base, Thanos no longer needed his warship's travelling capabilities now he has the Space Stone as he can simply open portals to any place he desires, which he took full advantage of to travel unaided for the other Stones, first heading to Knowhere for the Reality Stone and then taking Gamora with him to Vormir for the Soul Stone.
Thanos then made his way to his homeworld where he was confronted by the Avengers and Guardians of the Galaxy. Thanos made liberal use of the Space Stone during the battle first using it conjure a miniature singularity which absorbed Doctor Strange's attack before using the singularity as a weapon against the sorcerer. He later used it to harmlessly funnel the kinetic energy of Iron Man's projectiles before sending it back at the hero. In an ultimate display of the Space Stone's power, he used it to send the Power Stone's energies to Titan's moon, ripping it to shreds before using the Space Stone to send the chunks hurtling towards the planet's surface. After procuring the Time Stone, Thanos used the Space Stone to teleport himself directly to the Mind Stone's location in Wakanda.
Upon arrival, he used the Space Stone to phase the Hulkbuster armor through him and into a rock wall as well as knock down Falcon before using it to vaporize all of War Machine's bullets and then create compression forces around the War Machine armor, taking both of out the battle, as well as to defend from Scarlet Witch's stream of energy. After completing the Gauntlet and being stabbed by Stormbreaker, Thanos utilized all six Stones to wipe out half of all life in the universe before using the Space Stone one last time to teleport himself to an alien farmland.
As the Infinity Stone that represents and governs over space, the Space Stone grants the wielder absolute control over space itself. It is primarily used to open portals to other locations and can even allow interdimensional travel. As one of the six Infinity Stones, the Space Stone ranks as among the most powerful artifacts in the universe, with Black Widow saying it has the power to wipe out an entire planet. Like with all other Infinity Stones, it can only be used by beings of sufficient power and those who have no power are quickly consumed by it, although it seems to instead transport them rather than killing them unlike other stones, as evidenced with the Red Skull being quickly consumed by it when he touched it and was transported to Vormir and transformed into the keeper of the Soul Stone. Steel are quickly melted through by touch and specialized gloves will only keep the Stone from consuming any human bearer for seconds before killing them, with it still being painful.
Drawing power from space, the Space Stone's source of energy is so great that many consider it's volumes of energy to be unlimited. Due to this, the Space Stone is capable of powering or repairing objects and artifacts of incredible power, such as the Bifrost Bridge. A single spark of its power was enough to create a small explosion. HYDRA was able to use the Space Stone's energy within the Tesseract to power advanced vehicles and weapons of mass destruction including guns, tanks, and bombs that vaporized their targets. The Space Stone also emits low levels of Gamma Radiation when inside the Tesseract, which interfered with Selvig's methods of harnessing its energy source, and later allowed the Avengers to track its movements. The Space Stone, while an inanimate object, seems to be hold some measure of intelligence or temperament, since when Erik Selvig was studying it, he noted that it "behaves". When they turn off power used to harness its energy, the Space Stone turns the power back on using its energy source. Wherever the Space Stone is drawing its cosmic power from, it appears to be able to be manipulated from the other end point as well.
After Thanos inserts it into the Infinity Gauntlet, it is shown that if wielded by someone who truly understands and masters its powers, the Stone's control over space can be used to gain a wide range of potent and dangerous capabilities, not limited to simple teleportation, with Thanos displaying that with his strong understanding and mastery over the Space Stone's powers, he is able to use the Stone liberally in battle to great effect, warping the fabric of space to achieve a variety of powerful effects. Even when using teleportation, Thanos displayed a highly precise control over his portals, to the point where he was capable of using the Stone to selectively control what objects are transported, even removing the impaled Stormbreaker from his chest upon teleporting away, as well as being able to decide the size of the portals to allow him to teleport himself and multiple individuals, as shown when he created a portal large enough to warp both him and his children out of the Statesman. Aside from the ability to create portals and wormholes throughout the Universe, Thanos was able to skilfully use the Stone as a powerful form of telekinesis, warping the fabric of space around objects in order to manipulate and move them with just a single gesture of the Gauntlet. This ability proved to be among Thanos's most often used. The Stone was able to instantly freeze Loki's knive mid-stab with a casual gesture, effortlessly crush the guns on War Machine's armor, crush the wings on the Exo-7 Falcon to knock him down. Such was its strength, the Stone was even able to telekinetically pull the shattered fragments of Titan's moon across space to rain down on the attacking Avengers and Guardians of the Galaxy during the Battle of Titan.
Thanos also displays that the Space Stone can be used to achieve far more varied capabilities in combat apart from telekinesis. Upon shattering a gateway to the Mirror Dimension thrown by Doctor Strange, Thanos used the Stone to generate powerful compression forces to compress the gateway's fragments into a destructive black hole to throw back at the sorcerer. Thanos even had the skill needed to use the Space Stone to phase and unphase objects out of space, rendering them incorporeal and just as easily make them corporeal again, such as when Thanos used the Stone to phase the attacking Hulkbuster out of space, causing the Avenger to fall through the Titan harmlessly and fall into a cliff and then make him corporeal again to trap him in solid stone. Thanos shows the ability to use the Space Stone to create extremely strong force-fields of blue energy capable of instantly vaporizing the bullets of War Machine's armor and even blocking the energy of Scarlet Witch.
Thanos also displays the ability to combine the Space Stone with the other Stones to enhance it's effects and make it capable of a larger variety of feats. Combined with the Power Stone, Thanos can even gain the ability to warp and manipulate energy, using the Gauntlet to absorb the explosive energy of Iron Man's missiles, and redirect it at the Avenger as a single concentrated beam of fiery energy. He was also able to combine the Power Stone and the Space Stone to send the Power Stone's energies to Titan's moon by opening a portal to it. Together with the Reality Stone, Thanos shows the power to manipulate the terrain at such a scale as of to pull the surface of Titan itself towards him in order to grab the mid-air Doctor Strange. 
- In the comics, the Space Gem was a purple Infinity Gem that increases the speed of its user and allows them to be teleported wherever they want. It was shattered alongside four of the other Infinity Gems, aside from the Time Stone, when the Illuminati tried to stop another universe from colliding with their own in the cosmic event of Secret Wars (2015). After the restoration of the universe, the Infinity Gems were recreated, with the Space Gem now being blue.