Continuing a bit off of this question, at the end Mockingjay, the final book of the Hunger Games saga, a certain character ends up dead. The reason for the character's death is never fully explained.
Specifically, Katniss kills President Coin when Katniss is given the task of killing the ex-president Snow. Katniss subsequently gets arrested and trapped in a room for a few days while other events happen. Why does Katniss kill President Coin, when she agreed to a new Hunger Games shortly before?
According to Snow (and Katniss chose to believe him), President Coin ordered the attack on the Capitol, and in particular the bombs placed in the supply packages. One of these bombs killed Prim.
At a more fundamental level, it's because Katniss believes that Coin is as dangerous as Snow - or more dangerous, because Coin will lie and Snow didn't.
This is the same reason she rejects Gale in favor of Peeta.
What no-one here has explained is Katniss' reason for believing what Snow tells her. His logic is impeccable.
The bomb that kills Prim was delivered by a flyer. Snow told Katniss that given how close the rebels were to taking his home, if he had a working flyer at his command, he would be using it to get out of there, not blowing up people he still viewed as his subjects.
Then how the bombs were delivered, using the 'tribute gift' parachutes from the Hunger Games so the kids would go and pick them up. That takes forethought and planning, and Snow's troops were fighting to protect the capital. This part of my answer is speculation, and not confirmed in the books.
Finally, Coin was manipulating Katniss from the moment they met.
All of this told Katniss that Coin would be just as bad a president as Snow.
First and foremost, anyone thinking critically while reading Mockingjay or after reading it would simply need to recall the formation of the rebellion that takes place in the book(s) rather than the events that happen toward the end of it. President Coin declared herself leader of the REBELLION despite the obvious fact that it was KATNISS who ignited, fueled, and led it.
Second, Coin never gave Katniss control over her image as the Mockingjay--even when Katniss acted on her own accord, Coin ALWAYS fabricated Katniss to suit HER intentions for the rebellion. It doesn't take much to realize that all Coin wanted was authority--I wasn't even surprised when Katniss killed her at the end because of the various allusions to her tyrannical dark side that were so subtle but "out there" in her personality, especially when her and Katniss interacted (it was never pleasant).
In essence, Katniss and Coin did not see eye-to-eye regarding the rebellion. Katniss' lack of conformity was the distinguishing factor. This is why Coin wanted Peeta rescued at the end of Catching Fire because he conformed to the Capitol and made his and Katniss's affair seem convincing--Katniss couldn't achieve that, and President Snow threatened her for it. Even Haymitch admitted that Peeta was more simple to be around than Katniss because he could be persuaded and make things go smoothly (Remember the events at the end of The Hunger Games and during the first half of Catching Fire?). There is even a moment in the novel during a broadcast in which Peeta, still his normal self, attempts to get through to Katniss by telling her that she is a pawn being used to fulfill another regime. Boggs even told Katniss that Coin (without specifically saying it) was afraid that Katniss would rise to power on her own without even trying. To counter Katniss, Coin performed numerous tasks to suppress her including sending Peeta with Squad 451 in the hopes that he would kill her in a fit of unexpected rage.
The bombing of the Capitol children and Prim wasn't to break the last of the allegiance the people of the Capitol felt to Snow, but to mentally unhinge Katniss so that she would place her entire faith in Coin. And Coin nearly succeeded given that Katniss's depression and desire for vengeance briefly consumed her. Snow's last conversation didn't even need to be forced so that he could get through to Katniss; he knew how intelligent she was. All she had to do was put the pieces together to understand just how much worse Coin was in contrast to Snow. But even then she was trying to find a way to deny the truth because she was overwhelmed with sadness and anger...so when Coin suggested a "final" Hunger Games using the children of the Capitol's elite families, it solidified the truth in its entirety.
Coin made everyone fight a battle only to repeat an aspect of an atrocity they originally rose up against. A sane person like Katniss, Beetee, Haymitch, Peeta, and Annie knew it wasn't right despite the fact that the people of the CAPITOL enthusiastically watched the Hunger Games each year with no objection. Coin's plan wouldn't pacify those who directly suffered despite her claims that it would--there would never be any true compensation for the those who were lost in the games, the first war, and the second war. And, had a Hunger Games been hosted with Capitol children, Coin would have likely used another excuse, that the people were seeking more vengeance...it would eventually paved the way for her to embody and take over Snow's position. There would have been no change.
Katniss voted in favor of the games for one reason, and that was to gain Coin's trust. Her bargaining for the role as Snow's executor was simply a facade to cover up her own intentions. Katniss only had one chance to kill Coin and if she ruined that in any way...nothing would have changed.
It seemed pretty obvious that Snow wouldn't have bombed the people at his gates that were stopping the rebels from entering. Didn't take much logic to see that. Perhaps the most interesting character (and most difficult to figure out) is Plutarch. Who's side was he on?
In his final letter to Kat, you see that Plutarch was the real mastermind. He was the game maker. He brought in Kat despite Coin's reservations. As the person in charge of optics, he would have been the one to stage the Snow execution to have Kat lined up perfectly to shoot Coin instead. Until that moment, it seems that Kat had not decided to kill Coin. Plutarch, knowing what Coin was, positioned Kat in a place where he hoped (and expected) her to kill Coin, allowing for true, free elections. In Plutarch's final letter, he told Kat that she was exactly who he thought she would be.
Katniss killed Coin because she believed the Coin was dangerous, and was responsible for Primrose's death. Why does she believe these things?
Katniss's little sister, Primrose (nicknamed Prim), was killed in an explosion outside of the President house, along with hundreds of other children. Before what was supposed to be President Snow's execution, Snow told Katniss that Coin had ordered the attack on the kids, and since Prim was the reason Katniss had been in the games in the first place, she was devastated. So, to get revenge, Katniss shot President Coin, also allowing for Paylor to become president.
I think that Katniss believed Snow, as we all know Snow was a lying weasel always thinking for himself, I too believed that Snow would have escaped in a hovercraft if he had one. So she killed coin for all the deaths of the children. Also coin had killed Prim intentionally as Prim had been too young to be in a rescue mission so she had to have had Coin's permission to be there.
Okay basically this will take some time to explain, so I'll do it step by step.
#1: Gale and Beetee design a top-secret bomb that will blow up twice - one time for enemies, another time for healers
#2: At the capitol, a hovercraft with the symbol of capitol drops bombs that blow up twice (shocked expression)
#3: Katniss's sister, Prim, dies when the bombs blow up a second time
#4: Katniss talks to Snow, who says that they promised not to lie to each other
#5: Snow says that the Capitol DID NOT drop the bombs and that the bombs removed all the people of the Capitol's trust in him, something he would never do; It is also implied that President Snow has no hovercrafts left, and if he did, he would flee with them
#6: Katniss realizes that President Coin has been playing with her and Snow (Snow thinks Katniss is the enemy, Katniss thinks Snow is the enemy)
#7: Katniss learns that President Coin plans to "take President Snow's place", basically becoming President Snow again
#8: President Coin wants to hold a new type of Hunger Games
#9: Katniss realizes that President Snow is right and that President Coin is lying to her and killed Prim (on purpose, the main reason being to make sure that Katniss kills Snow; this move also removes any questions Katniss may have for Coin, but Snow foiled that part by telling Katniss; Also, President Coin's plan has too many loopholes)
#10: Katniss takes revenge for Prim and kills Coin to protect the children of the Capitol from the Games
