The basic victory condition is simple enough: Destroy colonies and colony ships. But the AI can also surrender early and be defeated that way. So it sounds like the Advent was very stubborn, but the Vasari decided to surrender early.

You don't get any special message for surrender, just says So-and-so was defeated.

In the cases I've seen, the game sometimes disregards the colony ships held by AIs who have lost all their colonies, so even after being declared defeated, I've stumbled across a planet they recolonized.

There's still one more detail in victory conditions: whenever you are involved in an alliance (only when diplomacy is enabled), victory is achieved only when all factions hostile to you and to your allies are destroyed or surrender. If the game does not end even though you have allied with all the other AIs, the game has not ended because there are at least two of them that hate each other, preventing victory from being achieved.

Interesting. I was playing a 4 player single-player game and I was allied to one of the computer players. As soon as my ally and I destroyed the last of the other 2 players, my ally canceled everything and went hostile on me.

I believe it was because you only had a cease-fire, not a peace treaty. 

Peace, Obi 1

yeah, but then you've already won!?  : )

After an enemy surrenders, their remaining planets can be in rebellion.  If you look at the forces, they will normally say rebel force xxxx rather than the old ai name, like in the initial stages of the game.  Except that there can be one huge fleet of rebels with advanced ships.

pek

What is victory?

Lets take a lecture from the teachings of the great Star Trek Villian, Gul Dukat

"A true victory is to make your enemies see they were wrong to oppose you in the first place, to force them to acknowledge your greatness"

In sins its slightly different, just wipe out everything and anything that opposses you!

To crush your enemies, see their fleets driven before you, and to hear the lamentation of the noobies.

Anyone want to hear my defenition of modern law? of course you do:

A system of respectively flexible, judicial and moral values structured as a semi-codified programme specifically idealised by a credible governing authority and designated by a practically recognised central administration to manage a congregated society.