Charter Day is the anniversary of Columbia's "official" founding on October 31, 1754, the date on which acting Governor DeLancey signed a royal charter for King's College on behalf of King George II. Yes, Columbia was founded on Halloween. Although Columbia operates under a different charter today, the anniversary of the 1754 royal charter is considered Columbia's birthday.
At various points in Columbia's history Charter Day was celebrated with an annual dinner, although funding was struck from the University budget in 1970, and it probably died out shortly thereafter. Curiously, Giving Day, the annual one-day online fundraising event held towards the end of each October since 2012, has no tie-ins to Columbia's birthday, though such a connection might otherwise seem obvious.
Alternatively, Charter Day can be April 13, 1787, the date on which the New York State legislature passed the charter constituting Columbia University (then just "Columbia College") as a private institution. In fact, there was even a Columbia College Charter Bicentennial held in 1987, although students were kind of apathetic about it, except for the ones who held protests.