Ever since she won The Hunger Games, Katniss Everdeen (Jennifer Lawrence) had an ongoing rivalry with the sadistic leader of Panem. President Coriolanus Snow (Donald Sutherland) saw Katniss as a threat for defying the rules of the games, and she proved him right by becoming the face of the rebellion against the Capitol.
Snow was set up as the big bad throughout the series, yet when Katniss was given the chance to kill him, she instead shot her arrow at President Coin (Julianne Moore). The Hunger Games: Mockingjay - Part 2 didn't explicitly state why Katniss killed Coin instead of Snow, but there were some clues along the way that explained her decision.
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Introduced in Hunger Games: Mockingjay - Part 1, President Alma Coin was the leader of District 13, a secret colony located near District 12. She was a strict, no-nonsense leader who was willing to let people die so that her mission could be fulfilled. Katniss and Coin didn't get along during their time in District 13 because Coin didn't believe that Katniss had what it took to lead a rebellion. Coin appeared to be a fervent supporter of the rebellion, but she took advantage of the movement to further her own political interests.
Katniss was determined to kill President Snow in Hunger Games: Mockingjay - Part 2 and led a team of rebels into the Capitol to find him. She and Gale were able to make it as far as Snow's mansion, where Snow had announced that children would be allowed in for safety. As Katniss neared the entrance, a hovercraft flew overhead and dropped gift packages into the crowd, but they turned out to be bombs that exploded and killed people in the surrounding area. A group of rebel medics, which included Katniss' sister Prim, went into the crowd to help survivors, but they were killed by a second bombing.
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Katniss went to see President Snow while he was imprisoned, and he revealed that he didn't order the bombing outside his mansion. He deduced that Coin had sent the bombs in aircraft emblazoned with the Capitol seal to turn the last of his supporters against him. Katniss also suspected that Coin deliberately sent Prim to the frontlines with the hopes that her death would push Katniss over the edge and make her an unsuitable choice for a leader.
Near the end of Mockingjay - Part 2, President Coin gathered the surviving victors for a meeting to discuss the future of Panem. She announced that she would be stepping in as interim president until the country was ready to hold an election. She also suggested that they hold a symbolic Hunger Games with the children of Snow's allies instead of executing hundreds of Capitol citizens. Although Katniss knew the turmoil that came with being a player in the games, she voted in favor of the symbolic games. She claimed that she wanted to avenge Prim, but her decision also allowed her to gain Coin's trust.
Katniss could see that Coin was already showing signs of being as ruthless as Snow, which could be dangerous for the future of Panem. Coin was willing to sacrifice children to turn the Capitol against Snow and make herself look like the hero that swooped in to save Panem. This is why Katniss chose to kill Coin at the execution instead, putting an end to her reign of terror before it could begin. Katniss would never agree to a president that could justify killing children. More importantly, she couldn't forgive Coin for planning an attack that got her sister killed, especially after she worked so hard to keep Prim safe.