From the Wikipedia article on the castle :
Barnard Castle (grid reference NZ04911641) is a ruined medieval castle situated in the town of the same name in County Durham. It is a Scheduled Ancient Monument, and was designated as a Grade I listed building in 1950. The remains of the medieval chapel of St Margaret in the outer ward are listed as Grade II.
A stone castle was built on the site of an earlier defended position from around 1095 to 1125 by Guy de Balliol. Between 1125 and 1185 his nephew Bernard de Balliol and his son Bernard II extended the building.
In 1216 the castle was besieged by Alexander II, King of Scotland. It was still held by the Balliol family although its ownership was disputed by the Bishops of Durham. When John Balliol was deposed as King of Scotland in 1296 the castle was passed to the Bishop of Durham. Around 1300 Edward I granted it to the Earl of Warwick. In the 15th century the castle passed by marriage to the Neville family. In 1477 during the Wars of the Roses, Richard, Duke of Gloucester (later Richard III) took possession of the castle, which became one of his favourite residences.
Over the next two centuries the Nevilles enlarged and improved the estate and created a substantial and impressive castle. However when Charles Neville, 6th Earl of Westmorland was attainted for his leading role in the Rising of the North the Neville estates were sequestered. In 1626 the Crown sold the castle and also the Neville property at Raby Castle to Sir Henry Vane.
Vane decided to make Raby his principal residence and Barnard Castle was abandoned and its contents and much of its masonry was removed for the maintenance and improvement of Raby.
The castle is in the custody of English Heritage and is open to the public. Of particular interest are the ruins of the 12th-century cylindrical tower and the 14th-century Great Hall and Great Chamber.
From the longer Wikipedia article on the town
The castle was founded by the Normans shortly after the conquest, but enjoyed its heyday under Bernard de Bailliol during the latter half of the 12th century. The castle passed into the hands of the Balliol family (of which the Scottish king, John Balliol, was the most important member) and then into the possession of Richard Neville, Earl of Warwick. King Richard III inherited it through his wife, Anne Neville, but it fell into ruins in the century after his death.
Barnard Castle ("Barney") is a market town in Teesdale, County Durham, England. It is named after the castle around which it grew up. It is the main settlement in the Teesdale area, and is a popular tourist destination. The Bowes Museum has the best collection of European fine and decorative arts in the North of England, housed in a "magnificent" 19th-century French-style chateau. Its most famous exhibit is the 18th century Silver Swan automaton though art includes work by Goya and El Greco.
Barnard Castle sits on the north side of the River Tees, opposite Startforth and 21 miles (34 km) southwest of the county town of Durham.Nearby towns include Bishop Auckland to the north-east, Darlington to the east and Richmond to the south-east.
Barnard Castle's largest single employer is the pharmaceuticals giant GlaxoSmithKline, who have a manufacturing facility on the outskirts of town.