The White Guardian is one of the Guardians of Time, along with the Black Guardian and the color-coded New Paradigm Guardians added by the novels.
The Guardians control all of reality, personifying opposing concepts. And it's not at all racist that the White Guardian embodies good while the Black Guardian embodies evil. Even though the White Guardian looks like a Southern plantation owner in a straw hat sipping mint juleps. Even though he later tells Five that the lesser races... OK, I can't finish this paragraph.
Anyway, when the Doctor meets him later, the straw hat is gone. Instead, he wears a dead pigeon now. Dead pigeons are cool.
His main function in the universe is to send the Doctor on quests to find the Key to Time. He does this with Four and Romana in The Stones of Blood. In Big Finish Land, he sends Five and Amy (not that one) on another quest for it. In the DWM comics for the show's 30th anniversary special, he sends Seven, Ace, and Benny to find it yet again. He really needs one of those key locator things where you can just ask your phone where you left it. Or maybe he can fit Time with one of those electronic lock things so he doesn't even need a key, he can just push a button on his phone, like my car has.
But sometimes he does other things. Well, just one other thing, just one time. He makes a deal with the Black Guardian to hold a race for the Eternals to see who deserves Enlightenment and then cheats by getting the Doctor to interfere in the race so nobody wins. Because he is the very embodiment of fair play.
According to the novels, at the end of the universe, whoever currently holds the mantle of the Lords of Time gets to survive to the next universe and become the new Guardians. So Whitey was the previous universe's equivalent of Rassilon, which explains a lot. Bravo Hinton.