The duties of the Arbiter are to oversee the Rite after the death of the ruling Chancellor of the High Council of the Klingon Empire in the event that the chancellor dies without being challenged for his seat by another Klingon. The Arbiter is the one who also officially confirms who the new Chancellor will be after selection by the Klingon High Council.
Non-Klingons can be appointed to the position. This was done once with Captain Jean-Luc Picard being appointed to the position after the death of Chancellor K'mpec. (TNG episode: "Redemption, Part I")
If a Chancellor is challenged to combat by another Klingon, and is subsequently defeated, the office of Chancellor passes immediately to the victor, or to whoever the victor chooses to name as the new Chancellor. In that case, no Arbiter of Succession is appointed. (DS9 episode: "Tacking Into the Wind").