Edgar's coin toss is one of the more memorable and important scenes in Final Fantasy VI. It helped decide the fates of both brothers, leading Sabin to make a decision that would change both of their lives forever.
Sabin and Edgar's father had died. It was suspected that it was from poisoning by the Empire. Sabin took it a lot harder than anyone else. He was sad, and understandably so. More than that though, he was angry. Edgar chased him to the roof of the castle to comfort him, where Sabin had probably run to get away from everyone and everything to take in and emotionally deal with what had just happened. He wanted to mourn for his father. The people of Figaro didn't give him that opportunity, though. Almost immediately after Sabin and Edgar had heard the news of their father's death and escaped to the roof, a High Priestess went up and interrupted the brothers, letting them know that their father's last words were that he would entrust Figaro to the two of them.
I'm sure succession was a huge worry for the people of Figaro. Edgar and Sabin were twins, and if they battled for the throne, it could have really torn up Figaro and left the kingdom weak. Funny enough, though, neither one of them wanted the throne.
It made Sabin sick that succession was all anyone could think about after their father had died. He was upset that none of them were sad, and that none of them cared that his father might have been poisoned. I think in that moment he stopped caring for the kingdom. (More on that here.) Sabin didn't want to be a tool of the Empire like his father was.
Let's leave this place! Let's forget this crazy kingdom, and live our lives how we want to!
Edgar was more sensible at that time, and less emotionally driven. He reminded Sabin that it was their responsibility to take care of Figaro, and that it's what their father wanted of them. Even Sabin couldn't argue when his father's wishes were brought up.
Edgar proposed a coin toss. Neither one of them wanted to be King, but someone had to take care of Figaro. He said that the winner of the toss would get to choose whichever path he wanted for himself with no regrets and no hard feelings. Sabin would win if the coin landed on Heads. Edgar would win if it landed on Tails. It landed on heads, and Sabin chose his freedom.
Later on in the game on Setzer's airship, Setzer proposes marriage to Celes in exchange for him helping the party. Celes agrees, but with conditions, saying that it will be decided by a coin toss. If it lands on Heads, she will marry Setzer in exchange for his help, and if it lands on Tails Setzer will help for free. She borrows a coin from Edgar, and tosses it in the air. It lands on heads. Setzer picks up the coin and notices that it's double-headed. If you have Sabin in your party during this scene, he will notice the coin and remember the toss that gave him his freedom.
Edgar's coin was double-headed. Sabin was tricked.
I don't think Edgar was being malevolent when he tricked Sabin. I think he was actually being self-sacrificial. The deal was not whoever won the coin toss would be King. The deal was whoever won the coin toss would get to choose his own path. Edgar let Sabin choose what he wanted to do, keeping the responsibility of their Kingdom on his own shoulders. Edgar ended up taking the throne and giving up his own freedom so that Sabin wouldn't have to.