Roxas is Axel's best friend, at least according to our lovely pyromaniac. They probably first met when Roxas joined the Organization as number XIII. Axel soon to grew a liking to him, and the two became friends. The problem was, Roxas wanted something else. He wanted to find Sora his original. He figured that no one would miss him and decided to leave against Axel's desires. The leader of assassin Nobodies knew that if one betrayed the Organization, there would be no escaping their wrath. Besides, Axel would miss the only member he liked Roxas.
The duo's elements here are quite symbolic. Fire and light belong together. Where there is fire, there is light. But where there is light, there is not always fire. The connection lies within this. Axel is always chasing after either Roxas or Sora, because they make him feel like he has a heart. Roxas, however, feels no need to be near Axel, especially after he loses his memories. All he wants is to find the other half of his light Sora.
In their first fight, there is no real winner. Axel seems to enjoy fighting Roxas, and from that we can speculate that the two have sparred before. Axel's efforts to bring back Roxas are then interrupted by DiZ, who also wants Roxas for his own purposes. With both men yelling at him from either side, Roxas can't handle the pressure, and he calls out for his three friends until they go away.
With the next encounter, Axel mentions that he's been given orders to kill Roxas if Roxas does not return. Roxas then says, "We're...best friends, right?" Axel pauses, then replies, "Sure...but I'm not getting turned into a Dusk for..." Had he continued that sentence, he most likely would have ended it with "you." Now it seems that Axel doesn't care about Roxas, but that's not exactly true. Axel has his own goals. While he may be best friends with Roxas, he would not give up his life to save him, especially because Roxas doesn't remember a thing. Why would Axel betray the Organization for someone who doesn't even recognize him anymore? Practically speaking, it makes perfect sense. Their conversation continues on, and Axel realizes he has no other choice but to fight his best friend. Just as he is about to start the fight, DiZ freezes time and commands Roxas to go to the mansion.
They meet up again in the Basement of the mansion. At this point Axel is furious that Roxas won't remember him or return, so he follows his orders and attempts to kill him. Axel is defeated, and as he is fading away, some of the memories they had together return to Roxas. Axel says they'll "meet again in the next life", because he knows all too well that he'll be punished for failing to carry out his orders properly. Roxas states that he'll be waiting for that time, but Axel knows that only one of them is going to have a next life Roxas.