The Age of Ascension was a rite of passage in several cultures. It generally signified one's entrance into adulthood.
The Klingon Age of Ascension was the time when a Klingon child officially became a warrior.
The anniversary of the event was a day of ritual and celebration. On the tenth anniversary of Worf's Rite of Ascension (he apparently did it when he was 15) his crewmates aboard the USS Enterprise-D recreated the ceremony on the holodeck. (TNG: "The Icarus Factor")
The Ferengi Age of Ascension was the time when a male Ferengi was expected to leave his family and make profit on his own. In 2351, Quark left home immediately following his age of ascension. (DS9: "Family Business")