The White is the governing official of The Spectrum, which is turn is the governing body of the Seven Satrapies. Although the White thus holds tremendous power and influence, he/she is not a ruler, (that role being filled by the Prism, who in addition to being the religious leader is also the emperor of the Seven Satrapies.) Instead, the White is more analogous to a head of senate; one who oversees and directs a legislative body.
The White represents The Chromeria itself in Spectrum debates, and only votes in the Spectrum decisions in the event of a tie, which is uncommon, especially since the Spectrum consists of seven members, so whenever all members are present and none are abstaining, a majority will be reached.
It is unclear how much power the White is guaranteed to hold; Orea Pullawr and Karris Guile, the two Whites shown during the events of the series, are both forceful and charismatic individuals, so it is possible that their influence is due to their own personalities rather than the laws of the Chromeria.