The Vongola Ring Battles is Xanxus' first appearance in the anime and the manga. Although I have a preference for the manga, I have far more images for the anime. In all honesty, the two arcs follow pretty close between the two mediums, so we're good as gold. Now that that's out of the way. The Vongola rings are a set of seven rings based in the defining dying will elements. They are like the family treasure and the group of seven that inherit the rings are considered the heirs to the family. In this case, the battle over the rings shows the next group chosen to lead the Vongola Family. They can only be utilized by someone of Vongola blood and their guardians.
The Ring Battle is Xanxus second attempt at acquiring the family through force. The first was the Cradle Affair in which he directly challenged his father after finding out he was adopted. After he lost, he was frozen for eight years by the zero point breakthrough, a unique move that seals dying will flames. Xanxus uses the Vongola Rings as a challenge for the last candidate for the tenth boss position. He and his men fight Sawada Tsunayoshi and his friends for dominance. The truth of the battle is actually to kill his father and frame Tsuna for it. The last member of the Varia squad is a golem like creature called Gola-Mosca. It feeds on dying will energy. The Ninth is stashed safely in the suit and is eventually defeated by Tsuna. Upon Mosca's defeat and the revealing of the Ninth, Xanxus states that he will kill Tsuna as vengeance for his injured (and possibly dying) father.
Xanxus orchestrated this elaborate battle to not only kill his father, but wipe out every single piece of competition. It is my belief that if he had defeated Tsuna that the Vongola may have at least looked to him for leadership if the ninth had died since every other candidate was dead. At the very least it would be something. However, it was not meant to be! Enraged by Xanxus' ruthlessness Tsuna vows that Xanxus will never follow in the Ninth's footsteps. Xanxus and Tsuna face off, and Tsuna is named the victor after using the Zero-Point Breakthrough. Xanxus is frozen once more, he is thawed by the Vongola Rings, but the rings reject him. This makes the point of the ring battle absolutely moot as Xanxus would never be able to harness their power to inherit the family. At this point, Tsuna becomes the tenth boss candidate and his name is never to be spoken in Xanxus' presence ever again.