Arguably the most complete medieval bishop's palace in the country, the Bishop's Palace in Wells has been home to the Bishops of Bath and Wells for over 800 years. The current bishop only occupies one (mainly 19th century) wing, and much of the rest of the building is open to the public except when it is being used by the diocese.
For full details of the opening hours, admission prices, etc. please see the official site, detailed below.
If nothing else, it is worth visiting the palace gardens to see the St Andrew's Wells, from which the town gets its name. These are the exits from an underground cave systems that starts in the Mendip Hills and brings water bubbling up into the lakes in the palace gardens. The view of the Cathedral reflected in the lake is a classic scene and not to be missed.