Two-Face's Coin

Two-Face uses a two-headed silver dollar, one side of which was scarred, the other side clean. Two-Face dictates any significant life decisions by flipping this coin.

If the coin lands on the scarred side, Two-Face will pursue acts of evil. If the coin lands on the unmarked side, he is compelled to commit acts of good. Two-Face often defers to his coin in choices of life and death.

Before Arkham Origins Incident
Harvey Dent's father was an alcoholic who would regularly beat his son. Every time, he gave his son a "chance". If the coin landed on heads, he would beat him. If it landed on tails, he wouldn't lay a hand on him. Little did Harvey know, the coin was double headed. So, Harvey was destined to lose every single time. When his father died, he inherited the coin and finally found out why he always lost.

Dent became a lawyer many years later, eventually joining the Gotham City District Attorney's Office. He would always keep the coin on his person as a reminder to stop men like his father, who was never prosecuted. Dent was elected D.A. after earning a sterling reputation as a "White Knight" of justice.

Before Assault on Arkham Incident
Dent worked with James Gordon and the vigilante Batman to take down organized crime in Gotham. Their first target was the Falcone Crime Family and their boss Carmine Falcone. Carmine was arrested and Dent was selected to prosecute. He had his coin in his coat pocket when he entered the courtroom. As Dent presented his case, Carmine took out a vial of acid that one of his men smuggled into his cell. Falcone threw the acid on the District Attorney, scarring half of his body and one side of the coin. Whenever the coin landed on the unmarked side, the Harvey personality would come out and be compassionate. Whenever it landed on the scarred side, the Two-Face personality would come out and be cruel.

Arkham City Incident
By this point, Two-Face was completely dependent on the coin for any of his criminal acts. Hugo Strange took advantage of this and tested his theory on whether or not madness could be cured. He switched Dent's coin with a copy to ensure Harvey did what he wanted. This would mean that Two-Face made a decision by himself. When he revealed this to Two-Face, the former attorney had a breakdown trying to find the difference between the coins, but he could not. Strange gave Dent a choice as he flipped the real coin: let the coin fall and be given therapy or catch the coin and find out what Catwoman was up to. He chose to catch it and Strange told him Catwoman was breaking into the D.A.'s old campaign office, where Two-Face keeps his valuables.

Two-Face captured the burglar and took her to the Solomon Wayne Courthouse. The two personalities came into conflict on whether or not to give her a trial or simply execute her. They decided to let the coin decide. Unmarked, there would be a trial.... Scarred side, she would get a bullet. It landed on heads and the court was now in session, much to the dismay of his henchmen. Batman intervened and was promptly shot by the attorney. He flipped his coin to either free her or kill her. The coin landed on on the scarred side and as he was about to shoot her Catwoman managed to break free from her ropes and scratched him, allowing Batman to recover and tie up the D.A.

Arkham Knight Incident
The coin was mentioned by Alfred when revealing to Batman that Two-Face was robbing banks, where he humorously speculated that Two-Face had misplaced it as a potential motive. During the Two-Faced Bandit Most Wanted Mission, every time Batman took a henchman out, Two-Face tossed the coin and informed his henchmen whether the family of the unconscious thug would get money or a bullet. When Batman stopped the robbery at the third bank, Two-Face himself arrived and flipped his coin. It fell on the scarred side, so he told his men to show no mercy.

When Two-Face was arrested, his belongings, including his coin, were placed in evidence storage.

A Flip of the Coin Incident
After reaching Two-Face, Robin was shot on the foot by the man himself. However, due to both Oracle using the PA system and Two-Face throwing his coin to determine Robin's fate, he managed to take down Two-Face and grabbed the coin before it fell to the ground.

Trivia

 * On Xbox Avatar Marketplace, you could buy Two-Face's Coin.
 * Harvey being defeated after being distracted by his coin might be a small nod to Batman Forever, in which Batman defeated Two-face by throwing a handful of coins at the same time as Harvey threw his, confusing him and causing him to lose his balance and fall to his death.
 * Although his father's abusive actions to Harvey Dent/Two-Face was implied with his mention that the coin was an inheritance from his father, it was never directly stated in any of the games.