What does Paul's "we will judge the world and the angels" mean?

Judges are not merely for the purpose of dispensing punishments, one tenet of being a judge is to restore order.

From wikipedia:

A biblical judge (Hebrew: shôphatîm or shoftim שופטים) is "a ruler or a military leader, as well as someone who presided over legal hearings."

"Legal hearings" could include sentencing punishment but also establishing one's innocence, and thereby attempting to reestablish the order to that of before an offense was commited.

And for Paul, we are more of partners with GOD recreating and redeeming the world. Of course He started that work and even is working through us as we become vessels of redemption in our own lives, but the passage is referencing how believers will be called upon to help restore order to this broken world, we will be vessels of reestablishing SHALOM.

NT scholar Craig Keener says:

in many Jewish traditions, the righteous would judge the nations; this judging could also imply judging the angels who were believed to rule the nations.

Colossians 1:19-20 ESV says...

¹⁹For in him all the fullness of God was pleased to dwell, ²⁰and through him God was pleased to reconcile to himself all things, whether on earth or in heaven, by making peace through the blood of his cross.

"Whether on earth or in heaven," seems to infer that there were things that needed reconciliation not only on earth, but also in heaven. That may be Paul's referencing how there may have been defective angels who needed reconciliation for their rebellion. However, we don't know much more about what Paul thinks of that because he doesn't make many mentions of such things.

Perhaps that is a roundabout way of saying that believers will be charged with the joy of reestablishing justice in heaven also!

Aside from that, the passage means exactly what you think it does!

For a good commentary on this see 1 + 2 Corinthians by Craig Keener.

I believe mature, spiritual Christians who know what the bible says are called to judge behavior accordingly. (See John 7:24, Galatians 6:15. 2 Corinthians 2:15, 2 John 10) There are so many examples in the bible where the early Christians had to resolve conflict in the church (thus judging a behavior) and deal with the perpetrators. I'm so tired of hearing people say "don't judge lest you be judged". This doesn't mean you can never judge someone's behavior. This is taken out of context because no one quotes the rest of the verses which states if you have sin in your life that is keeping you from walking righteously, you better not point out the splinter in your neighbor's eye. The bible is clear that h*m*sexuality is a sin, but people like Pope Francis an other compromising churches, don't want to judge. Well we are supposed to point out sins to others based on the word of God to help redirect them and put them on a path of righteousness.

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.