Guild Wars (Prophecies) is the original campaign, and should probably be played first, but the order isn't really important, as they are only loosely connected. Factions was the second campaign and is usually considered the worst, although it still adds some worth while content. Then comes Nightfall which has a much better pace, storyline and introduces heroes.
Any of them can be played by themselves as stand alone games, or can be linked to one another if you have more of them. Eye of the North is the only expansion and requires you to own one of the other games.
Correct. Eye of the North is the only true expansion, requiring any of the three previous games to access. You don't have to get all of the campaigns, although adding them to your account means you have access to more content.