V. Of the Holy Ghost
The Holy Ghost, proceeding from the Father and the Son, is of one substance, majesty, and glory, with the Father and the Son, very and eternal God.
This seems on the surface pretty straight forward. Some Anabaptists at the time were teaching that the Holy Ghost was not a distinct person of the Trinity, as is the Son or the Father. This has led many historians to believe that this is a clear answer to those kinds of people who believed that the Holy Ghost was some form of God the Father in action and not a distinct person of the Trinity. Anglicanism is extremely Trinitarian, so this is not a surprise that such an elaboration on the Holy Ghost is considered foundational and primary in Classical Anglicanism.
It should be noted that this article contains the Filioque clause, which has the Holy Spirit proceeding from the Father and the Son. This makes our Eastern Orthodox brethren go absolutely crazy. This phrase was not found in any of the original versions of the Creed that actually were approved at the Nicene Council. The West did not start using the phrase until the Provincial Council of Toledo in the 6th Century. Charlemagne was a big proponent of it, though the Pope at the time refused to add it to the Creed over Charlemagne's objections. The phrase was largely unused by Rome inself until AD 1040 when it seems to have been adopted into the language of the Pontifical mass for reasons that remain unclear. The Lambeth Council in 1978 and the Episcopal Church in 1994 have both passed resolutions agreeing that any future versions of the Book of Common Prayer will omit the Filioque clause. How Anglican churches that view or are mandated by Church law to follow the 39 Articles as a central Confession of Faith and therefore binding as doctrine will be hard pressed to follow those resolutions unless they want to alter the 39 articles as well. Lambeth has not spoken as to how that will be accomplished without violating one or the other.
One will also note that this Article defines the nature and person of the Holy Spirit, but, like the Creeds, the article is extremely vague as to what exactly the Holy Spirit's role is in relation to the Church or Individuals. The first 5 articles, though, are about the centrality and unity of the Trinity, not necessarily their particular functions.