Applejack is obviously Southern, and Rainbow Dash is the element of loyalty, so she is Union.
Being so full of herself, Rainbow Dash expected to win quickly as soon as the brawl broke out. However, Applejack managed to deliver a stunning kick right in Dash's jaw. The pegasus now knew she was in for a long fight.
Since Applejack was naturally a good kicker from years of apple-bucking, she scored several early blows on Rainbow Dash. The fight appeared to be going in her favor, but it was not long until Rainbow Dash's ability to fly evened things out. Applejack was only tiring herself in her attempts to strike her airborne foe; it had then become a battle of attrition.
Meanwhile, Twilight and Rarity cheered Applejack on after witnessing her early success. They had even considered assisting Applejack in the hopes that a defeat would humble the boastful Rainbow Dash. However, they decided against that notion after witnessing the pegasus retaliate with devastating results.
It was apparent that Applejack was fighting a losing battle. After much bitter tussling, Dash went all out on Applejack, kicking her until she was unconscious, and several times more after that for good measure.
Upon coming to, Applejack was forced to swear never to fight Rainbow Dash again.
Story written by me, with major editing from Gearbox. [link]