Functionally speaking, the concept is quite simple. Any player that is allied with every other remaining empire in the game will be declared the winner (where 'allied' refers to either a 'Defensive Alliance' or a 'Full Alliance'). This condition is checked whenever a senate vote occurs, or a player is eliminated from the game. This means that merely having the alliance isn't sufficient, it has to last until one of those checks occurs.

Keep in mind that any empire may win this way, so it is vital that you keep an eye on which empires are allied with which other empires.