Up until near the end of season four we're led to believe Cersei or Jaime had Jon Arryn — previous Hand of the King — murdered to cover up Cersei's offspring being the product of incest, yet in the episode First of His Name
Lysa Arryn admits to poisoning Jon (her then husband) after being prodded to do so by Petyr Baelish, aka Littlefinger.
The prodder always has a reason for doing something even if it's rarely obvious. What purpose was served by killing Jon Arryn?
Petyr's endgame is complicated, and I'm not even sure we know what the whole plan is yet. He may not even know. Littlefinger's biggest skill is being adaptable: he knows how to take advantage of opportunities as they come up. It's entirely in character for him to create some chaos just to see what happens, knowing he'll make the best out of any outcomes.
However, there are two common threads that run through all of Petyr's plans, from the very beginning. Two things he wants more than anything else:
At the time the story starts, Baelish has gotten himself into a very key role in the Kingdom: Master of Coin. He did so largely through what, in modern times, we might call "creative accounting" -- lots of borrowing money to pay off loans, shuffling debt, etc. At this point, he's in very deep to the Lannisters, and has basically been stuck in his current position for a while.
So, the status quo in the kingdom wasn't really working for Baelish, and he was looking for a way to stir things up. The opportunity came around when Jon Arryn discovered Cersei and Jaime's secret (which Baelish knew, because he had Jon followed). At that point, Baelish's plan started coming together. By telling Lyssa that Jon wanted to send her son off to be fostered with Stannis (for his own safety), and telling her that if Jon were gone he (Baelish) could marry her, he convinced her to murder her husband and, most importantly, blame it on the Lannisters.
His goal here appeared to include several possible elements:
It's even possible that civil war was always his end goal, much like it was Varys's, on the theory that he could do exactly what he eventually did: ingratiate himself to whoever seemed to be winning until he was able to "trade up" to someone with more power. Note that Petyr also quickly took advantage of Bran's attempted murder by trying to pin it on Tyrion, again knowing that it would drive a wedge between the Starks and Lannisters and weaken Robert's control over things. He also gave Lyssa strict orders to stay out of any political mess that might come up, knowing he wanted the Vale for himself later on.
We also certainly haven't seen the last of Baelish and his plans, so we may learn more in the future about his end game, and how much of it he had planned out all the way back when he was first plotting Jon Arryn's murder.
The answer is given by Varys in Season 3 Episode 4 when was talking with Lady Olenna Tyrell:
Lord Varys: Littlefinger is not long for the capital. A confidant of mine has told me that when he goes, Sansa Stark goes with him.
Lady Olenna Tyrell: And why have you come to me with this matter?
Lord Varys: Littlefinger was born with no lands, no wealth, no armies. He has acquired the first two. How long before he has the army? Perhaps you'll laugh, but I know him better than most, and this is the truth. Littlefinger is one of the most dangerous men in Westeros. If Robb Stark falls, Sansa Stark is the key to the North.
Lady Olenna Tyrell: And if Littlefinger marries her, he'll have the key in his pocket.
Lord Varys: Which seems such a shame. Why should a man with such a low reputation steal away such a lovely bride?
Lady Olenna Tyrell: You must despise him, you're working so hard to undermine him.
Lord Varys: Actually, I rather enjoy him. But he would see this country burn if he could be king of the ashes.
Littlefinger's end game is much more ambitious than most people realize.
Littlefinger was born the minor of the Lords, with just a few acres of lands on Fingers. However, he wants to rise up. But that's not plausible as Seven Kingdoms were dominated by following ancient Houses with big names and rich histories:
He wants to be the King but how must he do that unless there is a power vacuum? He explains his philosophy to Varys:
Chaos isn't a pit. Chaos is a ladder. Many who try to climb it fail and never get to try again. The fall breaks them. And some, are given a chance to climb. They refuse, they cling to the realm or the gods or love. Illusions. Only the ladder is real. The climb is all there is.
He wanted to create chaos which he would use as a ladder to climb up in social hierarchy. And of course he needed a Power Vacuum with waned power of the Great Houses, possibly extinction of some, so that their place could be filled by House Baelish. He does not care about the price, he would see the realm burn if he could be King of the ashes as Varys puts it.
So How did he plan to deal with the great Houses?
So Petyr had to kill Jon Arryn because:
As it happened, War took an interesting turn and Sansa Stark was left as sole heiress of Winterfell. With Vale under his control, Petyr Baelish believed it would be more beneficial for him to marry the heiress to the North and thus use the combined power of Riverlands, North and unscathed Vale to break the crumbling great Houses in aftermath of the War.
Petyr Explained his final motives in Season 4 Episode 4:
Petyr 'Littlefinger' Baelish: So many men, they risk so little. They spend their whole lives avoiding danger, and then they die. I'd risk everything to get what I want.
Sansa Stark: And what do you want?
Petyr 'Littlefinger' Baelish: [pauses] Everything.
He made it clear to Sansa in Season 6 Episode 10:
Every time I am faced with a decision I close my eyes and see the same picture. Whenever I consider an action I ask myself, 'Will this action help to make that picture a reality'. I pull it out of my mind and into the world, and I only act if the answer is yes. A picture of me on the Iron Throne, and you by my side.
And of course about his actions he had following thing to say:
A man with no motive is a man no one suspects. Always keep your foes confused.
Littlefinger had Lysa Arryn kill her husband, Jon Arryn -- Ned Stark's friend, who had been investigating the parentage of the incest-made Lannister children.
Littlefinger implicated Tyrion in the knife carrying, and in the books he did so knowing full well that it was actually Joffrey in order to cause more chaos. As noted above, the person who hired and armed the assassin in the books was Joffrey -- and Littlefinger knew it. In order to impress Robert (who said putting Bran out of his misery would be the right thing to do), Joffrey stole the dagger and hired the assassin.
In the show, however, it is revealed in the Season 7 finale that the knife was Littlefinger's own -- Bran reveals that the knife belongs to him at his trial, which allows Sansa to draw out the implication that the enmity between the Starks and the Lannisters is due to Littlefinger's conniving.
These two threads work to pit the Lannisters and Starks against each other -- allowing Littlefinger to both dispose of one of the two people preventing him from separating Sansa Stark from her family and get himself one step closer to the throne, after betraying Ned and setting the events leading to the war in action.
With this knowledge, Littlefinger's chat with Varus becomes a tacit admission of his hand in starting the war by getting the balls rolling. He admits to wanting the throne and having a hand in Lysa Arryn's widowhood in one response while talking about the throne:
Lord Varys: The Lysa Arryn of chairs. Shame you had to settle for your second choice.
Petyr 'Littlefinger' Baelish: Early days, my friend. It is flattering really, you feeling such dread at the prospect of me getting what I want.
The assassination of Jon Arryn is one of the major reason that Ned Stark went to King's Landing, his suspicions against the Lannisters only further exacerbated by the assassination attempt on Bran.
Littlefinger flat out admits to stopping Sansa's marriage to Loras Tyrell:
Lord Varys: Thwarting you has never been my primary ambition, I promise you. Although, who doesn't like to see their friends fail now and then.
Petyr 'Littlefinger' Baelish: You're so right. For instance, when I thwarted your plan to give Sansa Stark to the Tyrells, if I'm going to be honest, I did feel an unmistakable sense of enjoyment there.
Sansa's being held captive was one of the major points leading to the war, and her marriage to the Tyrells was to be her way out of Lannister captivity.
Given our near-omniscient viewpoint, which Bran clearly has some insight into, we see that, yes, Littlefinger is absolutely at fault in starting the war -- and he did so purposefully, because chaos is a ladder and he intended to climb it to the top -- he wanted to be the king, and he plotted a way to eliminate everyone in his way.
So Jon Arryn would not discover that his own wife got pregnant by another man, just as he had discovered King Robert Baratheon's wife was impregnated by somebody other than King Robert.
There's a fan theory that Petyr Baelish is the secret father of Robert Arryn. Here's the support for that theory.
He warmly greets Robyn each time he sees him, and this is something he does not do for others. He's not affectionate with anyone else, but he hugs this young boy.
He once impregnated Lysa Tully, but forced her to drink an abortifacient poison to cause a miscarriage. (Which also removes Jon Arryn's doubts about whether his wife was fertile enough to get pregnant.)
Jon Arryn was nearly 70 when he died, and Lysa said he was too old to get her pregnant.
Baelish chose the mockingbird as his sigil, and the mockingbird is known for getting other birds to raise their eggs.
Robin's dark brown hair color matches that of Petyr Baelish, but does not match that of Jon Arryn. Jon Arryn was blonde as a young man.
It would be ironic if Jon Arryn discovered adultery in another marriage, but not in his own. If he could learn the truth about somebody else's wife, he could eventually learn the truth about his own wife. Baelish could not allow that to happen because Baelish would be one of the suspects, and would be killed by Jon Arryn.
There seems to be a lot of duplicate questions leading here, and it's nice to see how different answers change over time, based on what we see. Ultimately, the answer is-because Peter wanted the Iron Throne. And getting there as a commoner is virtually impossible. Time and again he had people killed or otherwise manipulated in an effort to further his own power. So in a very simple way-he was the guy who started all the wars.
