It is named so, because at the beginning of the battle the town's Sept tolled its bells to warn the citizens of the battle and to persuade them to stay inside their houses.
Robert Baratheon had been wounded and separated from Eddard Stark and the main body of the rebel army. He hid in the town of Stoney Sept while he healed from his injury. (It is implied that the whore Bella, whom Arya meets when travelling with the outlaws, is one of Robert's bastards.)
Jon Connington, then King Aerys's Hand, arrived in Stoney Sept with an army, having heard that Robert was hiding there. His troops began a systematic search of the town, but before they could find Robert, Eddard and Hoster Tully arrived with their own troops. When the fighting was joined, the Septons rang every bell in the town to warn the townspeople to hide, hence the name Battle of the Bells.
The battle raged throughout the streets and even rooftops of the town. Robert killed six men, including Rhaegar Targaryen's friend and former squire, Myles Mooton. Connington killed Ser Denys Arryn, a distant cousin of Lord Jon Arryn (who had also married Jon's niece), and also wounded Hoster Tully badly.
The rebels won in the end, however Connington was exiled by Aerys.