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 first Rite of Ascension had to be taken by the age of 13 in order for a Klingon youth to declare his intentions to become a warrior. (TNG: "Firstborn")
The final initiation ceremony involved the use of painstiks, as the celebrant walks along a line of warriors, intoning: "Today I am a warrior. I must show you my heart. I travel the River of Blood."
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")
In "Heart of Glory", Worf refers to the Age of Ascension as "the Age of Inclusion"; however, this is the only instance in which this name is used on-screen. That term was also used in the game Star Trek: Klingon to refer to the Age of Ascension.
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")