The Periphery refers to the region of space at the galactic rim.
The "Periphery" is more used for the part slightly in the direction of Trantor from the galactic core, where the planet Terminus is located. The term for the specific part of the Periphery on the far opposite side of the galaxy's disk from Terminus is the "Opposing Periphery".
Imperial control was weakest in the Periphery, as planet governors there frequently possessed their own fleets. Santanni revolted during Forward the Foundation. About 50 years after the First Foundation was established on Terminus, the emperor granted the Periphery autonomy. This effectually removed them completely from Imperial control, making the dissolution of the Empire far more apparent than it had been before. As time passed, more and more fringe worlds of the Galactic Empire were chipped away by the barbarian kingdoms of the Periphery.
Two centuries after the establishment of the First Foundation, even more of the Periphery had degenerated into barbarian kingdoms, and the Empire only controlled the inner third of the galaxy - but this inner third was always its core power-base, controlling three-quarters of the entire galaxy's wealth and population. As a result, many in the inner provinces still openly derided the suggestion that the Empire was in a "decline" at all.
In the years preceding the fall of Trantor, the Periphery became any area outside the Empire. As this area became larger, the Empire became less and less great. The political center of the galaxy shifted to the Foundation Federation, and thus to the Periphery.