The Three Virtures are what bound the BIONICLE Matoran together when they were on Metru Nui and Mata Nui. The three virtues are Unity, Duty, and Destiny. They started in Metru-Nui. When the Matoran relocated to Mata-Nui, the Matoran forgot the three virtues until the arrival of the Toa Mata.
Unity was the virtue most stressed, especially among Toa, as most Toa teams seemed to lack it. It stressed the need to work together and help others.
Duty stressed doing your everyday jobs dilligently and responsibly. A Toa's duty, for example, would be to protect the Matoran.
Destiny represented the ability to better yourself. It meant what a being was 'meant' to accomplish in their lifetime. Most beings had different destinies. For example, the Toa Mata/Toa Nuva were destined to awaken Mata Nui, while the Toa Inika/Toa Mahri were destined to save his life with the Kanohi Ignika.
Reason for existing
The partial reason for the existence of the three virtues is commonly believed to be to give the BIONICLE a moral to teach young children.
While the virtues were never explicitly mentioned again, the theme of unity, duty and destiny still comes up in storylines after Metru Nui.