A faction does not necessarily have to be something that you can join and climb in rank with. And the Elder Scrolls example that someone brought up is maybe not the best example of a faction system in my mind. I mean it's ok and all... but it lacks a huge level of depth, and what not, for me to think it is a proper faction system.
No... when I think about factions, and their effect on the player, I think about games like Freelancer (probably my alltime favorit space flight combat game to this date, my nr 2 is Tie Fighter).
Freelancer has almost 50 different factions, and every single ship and person you meet in the game is connected to one of these factions. Most of these factions are devided amongst 4 different nations (Liberty, Bretonia, Rheinland, Kusari), and the "Independents" (so not connected to a nation, like the "Bounty Hunters" for example)... there is also a a few factions which are listed as "Others" (which might, and might not, have some connection to other factions in the game).
Each nation consists of a few Government types of factions (things like Military and Police, etc), Corperations (manufacturing types, traders, etc), and Criminal/Pirate factions. The Independent consists of some seveal lawful factions (like the mentioned "Bounty Hunters"), a single neutral faction, and a few criminal factions. And the "Others" are... well... the others... XD
Your Reputation with these factions will determin how they react to you if they happend to come across you. Here is a fairly short
description on how your reputation level with a faction will work in Freelancer, especially the part under "Effects of Reputation" is worth reading since that tells you the effect of being "Friendly", "Neutral" or "Hostile" to a faction.
What you do in the game, and with/against who you do these things, will increase or lower your reputation with a faction (or might keep it the same, it depends). Not only that... what you do can effect your reputation with multiple factions at the same time. This specificly is the case when you have interactions with factions which are part of a nation. So if you for example do a mission for the Bretonian military faction, to maybe kill a certain pirate, then your repitation with the Bretonian military faction will increase a fair amount, and your reputation with all other Bretonian Government and Corporations will increase a bit to... but, your reputation with the targeted pirate faction will decrease a fair amount, and your reputation with the other Bretonian criminal/pirate factions will also decrease a little bit.
The same would happen if you decided to turn pirate and start robbing people, or doing missions for them once they are available to you... except reversed of course... where the government and corporations would think less and less of you, where as the criminals would think more and more of you. These things usually do not cross the boarders though... so the Rheinland's will not really care about what you do in other nations, unless if it where to directly effect them (not sure if there are side missions which would target another nations factions... I do know though that your reputation with some factions will be effected by doing the main storyline).
You can not actually join a faction in Freelancer, like how you can in the Elder Scrolls games... but that is ok... you are after all supposed to be a "Freelancer" in the game. XD
There are also a few other things I think is missing from Freelancers faction system... but all in all it is a pretty good one still.