Squire Asrol is found in the east end of the courtyard of White Knights' Castle. Speaking to him starts the quest The Knight's Sword, where he is in trouble and needs help from an adventurer. He explains that he has lost his master, Sir Vyvin's, sword. Since it's a very special ceremonial sword, he would be in big trouble if Sir Vyvin found out.
He also can be talked to claim the rewards of the hard and elite Falador Tasks after they are completed, and is an anagram in a level three clue, rewarding players that talk to him with puzzle boxes.