The Arbiter made his first appearance in Halo 2. Once a grand ship master, this elite was supposed to safeguard the sacred Halo ring, which was utterly destroyed by Spartan 117, the Master Chief, in Halo: CE. He was sent on a suicide mission to eliminate a heretic threat, that eventually resulted in the the Arbiter being the first Arbiter to live his mission. On his mission, he realized that "The Great Journey" is a lie, and the prophets knew it. After learning this, he and the other Elites broke away from the Covenant and allied themselves with the UNSC in hopes of defeating the Covenant and stopping The Great Journey.
The Arbiter plays a major role in Halo 2, and his story is interwoven into the main storyline of the Master Chief. When the Elite is turned Arbiter, you play as the Arbiter on his mission to eliminate the Heretic threat. The Arbiter has a deep and sonorous voice and speaks in a measured tone, giving him the air of someone very contemplative and thoughtful.
In Halo 3 the Arbiter returns as a playable character in Campaign Co-Op mode. The voice acting for the Arbiter in Halo 3 has changed somewhat, and during the Campaign he will make comments that sound more urgent than in Halo 2, diminishing the air of mystery and wisdom that was prevalent in Halo 2.
'Arbiter' is the title given to people in the Covenant who, instead of being imprisoned, are sent on a suicide mission. Past Arbiters lie in a Mausoleum, and during the cutscene where the grand ship master is turned Arbiter, we learn Arbiters are key to resolving Covenant issues such as events referred to as the Grunt Rebellion (uprising?) and others.
Becoming the Arbiter...
Arbiter: "What of the Council?"
"Your tasks as Arbiter will be arduous, suicidal. The council shall have its corpse."
On flight at start of Arbiter mission:
Elite captain: "Your armour suits you well, but it cannot hide that mark. My elites' lives matter to me, yours does not"
Arbiter: "That makes two of us".
Elite captain at the beginning of A Whisper in the Storm: "We are the arms of the Prophets, Arbiter, and you are their blade".