According to reformed theology, who or what will the saints reign over during the millenium? Regarding the 1000 years, Revelation 20:6 states:


  Blessed and holy is the one who takes part in the first resurrection.
  The second death has no power over them, but they will be priests of
  God and of Christ, and they will reign with him for a thousand years.
  (NET)


According to pre-millennial reformed theology, who or what will the saints reign over during the 1000 years?
While pre-millennialism is rare in Reformed theology, you will find it among the Reformed Baptists. It is possible that their views vary greatly today, I will assume that those of the eminent theologian John Gill (1697-1771) are still typical. As mentioned in the comments, this pre-millennialism is not dispensational, so do not confuse it with the popular modern versions.

Quoting from his Exposition of the Bible, on Revelations 20:4, as his commentary on verse 6 refers to the previous mention of "reigned with Christ a thousand years":


  ... Christ being descended from heaven, and having bound Satan, and the dead saints being raised, and the living ones changed, he will reign among them personally, visibly, and gloriously, and in the fullest manner; all the antichristian powers will be destroyed; Satan will be in close confinement; death, with respect to Christ and his people, will be no more; the heavens and the earth will be made new, and all things will be subject to him; and all his saints will be with him, and they shall reign with him; they shall be glorified together; they shall sit on the throne with him, have a crown of righteousness given them, and possess the kingdom appointed for them; they will reign over all their enemies; Satan will be bruised under their feet, being bound; the wicked will be shut up in hell, and neither will be able to give them any disturbance; and sin and death will be no more: this reign will not be in a sensual and carnal way, or lie in possessing worldly riches and honours, in eating and drinking, marrying, and giving in marriage; the saints will not be in a mortal, but in an immortal state; the children of this resurrection will be like the angels; and this reign will be on earth, (Revelation 5:10) the present earth will be burnt up, and a new one formed, in which these righteous persons will dwell, (2 Peter 3:13) of which (See Gill on Revelation 21:1) and it will last a thousand years; not distinct from, but the same with the thousand years in which Satan will be bound ...


I hope this summary is accurate, but from my reading of Gills Exposition, Reformed pre-millennialsm essentially is a belief in an extended "judgement day" that begins with Christ's second coming and ends with the final judgement, the millennium being a thousand years of glory for the saints before that final judgement of the wicked.

To get a clear picture of his teachings on the millennium and the second coming, I would recommend reading his full commentary on Revelations 20. Again, as a disclaimer, I only know one Reformed Baptist personally, and I have no idea if John Gill's pre-millennial theology is still popular among them.