A Powiat (pronounced povyat) is the second level of Polish administrative division, between Voivodships and Communes. As such it is roughly equivalent to counties or districts in the United Kingdom.
There is no official equivalent of the name in English.
Some of the towns in a Voivodship can be centres of more than one powiat. The reason is that in several cases the town itself and its surroundings form separate entities of administrative division. This is usually the case for large Polish cities, but can also apply to several smaller towns.
Powiat Zgorzelecki - Dolnośląskie - Poland Travel Guide