The Wandering Days is the term used by Hobbits to refer to the centuries-long period of their westward migration from their races' ancestral home somewhere in the wild East (probably in the upper vales of the river Anduin).
Over many generations the Hobbits drifted westward until many were settled between the Brandywine river and the Weather Hills, predominantly in the Bree-land. Eventually, in the year T.A. 1601 a large population of Hobbits was granted permission by King Argeleb II, one of the least few Kings of Arthedain to colonize the region which would become The Shire, the main homeland of most of Hobbit-kind.
The oral history and memory of the Hobbits does not extend further back than their "Wandering Days", with their exact origins having been lost.