Atium is a metal that has the allomantic effect to see into another person's future. Atium is the metallic form of the god Ruin, and is considered one of the two God metals, the other being Lerasium. Atium was originally thought to be the final, tenth metal and then it was later believed to be but one of the sixteen metals but both of these ideas were eventually disproved.
In Allomancy, Atium allows the user to see into another person's future. The user can see what their opponent is going to do before it happens, thus allowing them to avoid or counterattack any offense. This effectively renders the user invincible to any form of attack, unless their opponent is also burning Atium or Electrum. However, it has been proven that a person burning Atium can indeed be defeated by one not burning Atium nor Electrum, such as when Vin killed Zane. Furthermore, this ability is limited only to predicting the future centered around live beings, making natural events insusceptible to the perception that burning Atium grants
In Feruchemy, Atiumminds (pieces of Atium used for Feruchemy) store the attribute of age. When an Atiummind is being filled, the Feruchemist becomes older, and, by reversal, younger when it is being tapped. This Feruchemical property of Atium is what allowed the Lord Ruler to render himself immortal.
Through most of the books it is assumed that only Mistborn may burn Atium, as is the basic assumption of all metals above the eight basic. As such, the only strategies developed are by Mistborn using Atium and not so many concerning Mistings who can burn Atium, even though such Allomancers existed.
In general, Mistborns played a waiting game. This meant you hoped you had more Atium than the other Mistborn, or you could kill them before yours ran out. Throughout the books though, varying degrees of trickery are employed to circumvent any lack of Atium. In the first book, Mistborn: The Final Empire, Vin manages to fool Shan Elariel, another Mistborn, in a fight. Assuming that Shan, as a noble, had a greater supply of Atium, Vin extinguishes her metal before it runs out, creating the illusion that her Atium was gone. Because of this, Shan's self-confidence lulled her into a false sense of security. So when Vin burned the rest of her Atium, the confusion it caused in the other Mistborn gave her the opening required to deal the death blow.
Another instance where Vin uses no Atium to fool a Mistborn burning the metal is in the second book, The Well of Ascension, with Zane, as stated above. By reading Zane's body language, Vin correctly guessed what he saw, replicating the effect of her using Atium herself. Knowing where she would dodge, she was able to move the other direction. The split second of confusion again allowed Vin to finish the battle, killing Zane in the process.
Burning Duralumin with Atium has dubious value in combat, but it does appear to allow the user to see into higher dimensions for a short period of time. It is unknown if this also extends the amount of time the user can see into the future.
Atium was easily the most expensive object in the Final Empire. During the time of the Lord Ruler, Atium was kept under strict regulation by the Ministry. By controlling the mining through House Venture, the Lord Ruler controlled the most powerful of the metals Mistborn could burn. This allowed him a monopoly in the production of Atium, giving him an incredibly powerful source of income and a safety measure to control the nobility, in case of revolution or war.
Because the location of the Atium mining, the Pits of Hathsin a famously inescapable jail, nobody suspected that Atium was mined there, and therefore, a rebellion of Mistborn could be controlled. According to Kelsier and Dockson's research, the Lord Ruler only sold less than a tenth of the Atium that was mined. Through more reading, it is revealed he requires the Atium himself to stay young. This does not change the market situation. Keeping a strangle hold on Atium, and therefore Mistborn and the Great Houses they belonged too, the Lord Ruler was able to ensure a greater control over his nobles than through simple, though effective, fear.
Atium supplies were further limited by the Kandra. The Kandra only accepted atium in payment for their services as spies and imposters, and this payment was stockpiled, as per the First Contract between the Kandra and the Lord Ruler. Some of the atium produced may have been transported directly to the kandra, rather than in payment.
In the third book, it is revealed that there are in fact mistings that can burn Atium. This is found out after the mist snaps a large amount of people and some of them become "Seers" or Atium mistings. The reason that everyone thought that only mistborn could use Atium before was the fact that nobody was willing to waste Atium testing non-Mistborn children to check if they were Atium mistings, especially considering that nobody was even sure if an Atium misting could exist. However, the Steel Inquisitors secretly spiked nobles' drinks with atium during balls while the Lord Ruler was in power and caused a commotion to find Atium mistings, who would instinctively burn the metal. This was the method by which Yomen was discovered to be a Seer. Many other Atium mistings found this way were spiked to give Atium powers to Inquisitors.