Hi Barbara,
It's very confusing, but the Black Country is/was an industrial area centered around the iron and steel-making areas between Wolverhampton and Birmingham, so boundaries can be vague and there are a number of archives covering this region. County boundaries for Staffordshire, Warwickshire and Worcestershire, and now also West Midlands, cross through this area.
To make it even more confusing, a number of villages including Sedgley are located in Staffordshire but come under Dudley Registration District so it's necessary to check both Wolverhampton and Dudley registration districts for BMDs near to this boundary (for example, lots of people who were born or lived in what in practice were parts of Bilston were actually not inside the town boundary and officially in Sedgley (caused me quite a lot of confusion when I started researching).
Relevant archives include Wolverhampton, Dudley (at Coseley), Sandwell (at Smethwick), Walsall and possibly also Birmingham as there was a lot of interaction and movement between these areas. There are also the main Staffordshire County archives in Stafford and the Established Church archives at Lichfield. The archived catholic records for this whole area are at the Birmingham ArchDiocesan Archives.