The Citadel B ring is a section inside the Citadel.
The A ring continues to the B ring. Here are the barracks, the hospital, and "The Solar".
Knights and paladins rest in the barracks. Here they keep their belongings in the footlockers, and sleep on the beds (which seem in pristine condition, unlike the ones found in the courtyard).
The Solar is the Elder Owyn Lyons's personal quarters. It consists of a living room and a bedroom.
The Citadel B ring appears only in Fallout 3.