The ruined city near the Lonely Mountain later rebuilt by Bard is Dale. Dale was an old city of Men in the valley beside the River Running, close to Erebor and north of Lake-town.
Before the events of The Hobbit, Dale prospered through trade with the Dwarves of the Lonely Mountain. Its wealth and position made it a major northern centre, but the dragon Smaug attacked Erebor and destroyed Dale as well. The surviving people fled south toward the Long Lake, helping to form or strengthen the community that became Lake-town.
Bard the Bowman is descended from Girion, the last lord of Dale. During Smaug’s later attack on Lake-town, Bard kills the dragon with the Black Arrow. After the Battle of Five Armies and the death of Thorin Oakenshield, Bard receives a share of the recovered treasure. He uses that wealth to restore Dale and becomes its king.
Dale is therefore distinct from Lake-town: Dale is the old city rebuilt after Smaug’s defeat, while Lake-town is the waterborne settlement Smaug destroys during the same episode.