The Cylon transponder is a small, ovoid device that is discretely placed under Galactica's DRADIS console by a Cylon agent. A similar device was carried about by the Cylon agent, Number Six, during her time on Caprica with Gaius Baltar (Miniseries), perhaps to better allow tracking of herself by other Cylon agents.
The transponder works as an IFF device, identifying anyone or anything that holds it as a "friendly" contact to other Cylons. Apparently this device was installed by the first copy of Aaron Doral shortly before the Cylon attack began while security throughout the ship was more lax during the work of transforming Galactica into a museum. It was likely intended by the Cylons to track Galactica during (or, if she were to escape, after) the initial Cylon attacks.
There are two transponder styles. The second style is similarly shaped, but has cover plates that suggest advanced features, range, or programming.
Commander William Adama takes strategic advantage of one of the devices by placing it aboard a Raptor, allowing it to pass unchallenged into a Cylon basestar, where a nuclear device is placed aboard the basestar to destroy it (Kobol's Last Gleaming, Part II). A second transponder is aboard the captured Raider, piloted by Kara Thrace in her against-orders adventure back to Caprica in search of an ancient artifact (Kobol's Last Gleaming, Part I). This transponder is obtained from the copy of Leoben Conoy found on the Gemenon Traveller, in a scene cut from the episode "Flesh and Bone."
It is presumed that the transponder uses a form of wireless communication, although this has not been established in the series.
Keen eyes who watch the Miniseries will note that the transponder was not on the DRADIS console until later in the show, indicating that it was placed either shortly before or after the ship entered combat. This contradicts Gaeta's statement that "is has been there for about a week" and is likely a gaffe on the part of the producers.