What is The Second book of Homilies mentioned in the Anglican 39 articles of faith? In the 39 articles of faith, article 35 says the following:


  XXXV. Of the Homilies.  The Second Book of Homilies, the several
  titles whereof we have joined under this Article, doth contain a godly
  and wholesome Doctrine, and necessary for these times, as doth the
  former Book of Homilies, which were set forth in the time of Edward
  the Sixth; and therefore we judge them to be read in Churches by the
  Ministers, diligently and distinctly, that they may be understanded of
  the people. (Articles of Religion)


It then goes on to list the name of the homilies.

What is "The Second Book of Homilies" that it is referring to?
As per Wkipedia, the Second Book of Homilies is a book of twenty-one sermons written mainly by Bishop John Jewel and published in 1571. 


  Thomas Cranmer and other English reformers saw the need for local congregations to be taught Christian theology and practice. Before the English Reformation, the liturgy was conducted entirely in Latin, to which the common people listened passively [...]. Since parsons, vicars and curates often lacked the education and experience needed to write sermons and were often unfamiliar with Reformed doctrine, scholars and bishops wrote out a collection of sermons for them, which were appointed to be read each Sunday and holy day.


In short they were an exposition of Anglican doctrine intended for the general congregation. They are no longer read out, largely because Anglican clergy no longer "lack the education and experience needed to write sermons" and are not "often unfamiliar with [Anglican] doctrine".