What does Paul's "we will judge the world and the angels" mean? 1 Corinthians 6:1-4
1 When one of you has a dispute with another believer, how dare you file a lawsuit and ask a secular court to decide the matter instead of taking it to other believers! 2 Don’t you realize that someday we believers will judge the world? And since you are going to judge the world, can’t you decide even these little things among yourselves? 3 Don’t you realize that we will judge angels? So you should surely be able to resolve ordinary disputes in this life.

What does the passage above mean when it says &quot;believers will judge the world and the angels&quot;?
The book of Revelation refers to the final judgment (Rev. 20:11-13; NASB quoted): 


  Then I saw a great white throne and Him who sat upon it, from whose presence earth and heaven fled away, and no place was found for them. And I saw the dead, the great and the small, standing before the throne, and books were opened; and another book was opened, which is the book of life; and the dead were judged from the things which were written in the books, according to their deeds. And the sea gave up the dead which were in it, and death and Hades gave up the dead which were in them; and they were judged, every one of them according to their deeds.


At this point all are judged; that both men and angels are subject to this judgment is suggested by the Revelator saying that both earth and heaven try to flee from the presence of Him who is on the throne to judge.   (As to why angels would be judged -- recall that there are both the unfallen angels that serve God, and also the fallen angels that serve the devil; see, e.g. Rev. 12:7 where Michael and his angels battle the dragon and his angels.)

As regards who does the judging, certainly that belongs to Him who is on the throne, but a few verses earlier (Rev. 20:4) we also see others will as well:


  Then I saw thrones, and they sat on them, and judgment was given to them. And I saw the souls of those who had been beheaded because of their testimony of Jesus and because of the word of God, and those who had not worshiped the beast or his image, and had not received the mark on their forehead and on their hand; and they came to life and reigned with Christ for a thousand years.


I admit that the text isn't very clear on who the 'they' are that sit on the thrones here. Given, however, that the only other people mentioned in the chapter at this point are the angel who locks Satan in the abyss, and Satan himself, the implication appears to be that the martyrs and faithful mentioned immediately afterwards, reigning with Christ, are the ones sitting on the thrones and given the power of judgment. 

The beliefs of individual denominations about the further nature of the final judgment vary widely; the Wikipedia article describes several.  But in general this idea of the final judgment appears to be what Paul has in mind.