Jon Snow kinda sucks at war, and Davos Seaworth isn't much better.
So Snow has an outnumbered force, with basically no suitable weapons, and decides to meet Ramsay on the field. At the very least his idea of goading Ramsay was a good one. He even picked a decent defensive position, where the forests could have been used as cover against Bolton arrows and cavalry charge. Obviously the battle goes wrong, as he falls for Ramsay's calculated ploy to lure him into making a reckless and stupid charge. I'd easily argue that Bolton wouldn't have even needed to do it, but effective nonetheless.
So Jon loses his damned mind, and charges the Bolton cavalry by his lonesome. Davos sacrifices the linchpin in any maneuver the Snow forces hopes to make by sending out the cavalry to charge into the Bolton cavalry. Already the battle is destined to fail. Jon allows himself to get bogged down in the fighting, his forces engaged in messy and hectic melee. At this point he should have withdrew from battle, sounded the retreat, and reformed his lines. Ramsey's tactic of pelting the melee with arrows did work effectively to keep the Snow forces exactly where they were, however. Retreating forces would have taken an arrow to the back, but they would have had a better shot at surviving than being completely broken and helpless before the advancing, Bolton infantry.
You might think that the big, bad horses were the worst thing on the battlefield. Tormund comments that the enemy cavalry was what he feared the most on the battlefield, because how the wildling forces were cut down on the other side of the wall. What he failed to understand is that the most effective part of any army during this age is the infantry. In a shield wall formation the Bolton troops are damn near invincible from the front. The show aptly displayed how effective and deadly it can be. Even a veteran, skilled soldier has a poor chance of actually penetrating it-- Tormund got sliced up for trying. Jon's primary concern should have been these troops, and his entire strategy should have been about flanking them.
Then again, he probably went into battle with like no real intelligence on the Bolton formations.
Alright, so there's a horrible melee. While this was happening, Davos should have had some mind to divert some of his troops to flank around the enemy. At the very least it could have distracted Bolton's forces, or diverted some of the Bolton reserves away. Instead he orders a suicidal charge to join the fray. I can't blame him too much, because nothing would have been a super great idea at this point. No matter what he did, Bolton reserves would have swooped in to end the battle. Instead, Jon should have had the presence of mind to retreat as quickly as possible. Even tired his men would have been able to outrun the heavy, shield baring infantry that the Bolton's sent to EASILY take the rear of the broken Snow line.
If they could have regrouped in the forest, directly behind their initial point of deployment, they would have had a much better chance. The shield wall would have been less effective in the forest, especially with the Bolton cavalry out of commission. The Wildlings would have been more at home fighting among the trees as well. Would it have been a victory? I don't know, but it would have been a better idea than just slugging it out against a larger, better equipped force on the open field.
The entire battle was lost because of Jon. If it wasn't for Sansa's quick, desperate thinking then all would have been lost. Now Littlefinger is a punk-ass weasel, so I'm sure his help will come with a price at some point, but Sansa saved them all for now.