Bree-hobbits were the Hobbits native to Bree and the lands around; their interaction with Men (who also lived in the Bree-land), made them more open to the outside world than their parochial cousins in the Shire.
Bree itself was a very ancient township of Men. The first Hobbits came there from the east in about the year 1300 of the Third Age, making Bree the only place in the world where Hobbits and Men lived together side by side. Most of the Hobbits of the Bree-land lived in Staddle, a village on the southeastern slopes of the Bree-hill, but some also dwelt in Bree itself, living on the hill above the houses of Men.
It was three hundred years after Hobbits first arrived in Bree that two Fallohides, Marcho and Blanco, led a group of colonists westward to found the Shire. The Bree-hobbits, therefore, tended to look down somewhat on their cousins in the Shire, referring to them as 'Colonists' and 'Outsiders'. The Hobbits of Bree claimed to have originated most Hobbit customs, and certainly it was thought that they were the ones who first discovered the properties of pipe-weed.
At the end of the Third Age, the Bree-hobbits still occasionally travelled west to the Shire, but they rarely ventured further than Buckland or the Eastfarthing, those parts of the Shire that were closest to their home.
There were also hobbits by the name of Banks living in the Shire, and Underhill was a hamlet of Hobbiton, but we can do no more than speculate on the links between these families and regions.