To box a Cylon means to indefinitely download a particular 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, and is practically a form of permanent death. Boxing is done with a Cylon agent whose memories appear "corrupted" by extended or intense human contact. It is also done to sleeper agents who become psychologically maladjusted if their fictional persona tries to reject their Cylon nature.
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. Their "hero" status and extended human associations threaten the Cylon status quo because they had began to become individuals with differing personalities from the standard of their models (Downloaded). Incidentally, this also appears to be happening to the very model of Number Three who wanted to box Caprica-Six and Valerii (Hero).
When Gaius Baltar, held aboard a basestar, asks Caprica-Six why he hasn't seen the remaining five of the 12 Cylon agent models, she tells him that the subject is not to be discussed per Cylon directive, strongly suggesting that the other agents are either elsewhere, or boxed (Torn).