To box a Cylon means to permanently download a Cylon agent's consciousness from their body into cold storage (apparently a form of computer data dump).
To an agent, boxing is apparently a state of unconsciousness, almost a form of permanent death. Boxing is done with a Cylon agent whose memories appear "corrupted" by extended or intense human contact. The recovered Number Six known as Caprica Six (who died protecting Gaius Baltar when his house is destroyed in the Miniseries) and the copy of Sharon "Boomer" Valerii that had shot Commander Adama are threatened with boxing by a copy of Number Three shortly after their return, as their "hero" status and extended human associations threaten the Cylon status quo (Downloaded). It is currently unknown how many, if any, Cylons have actually been boxed.