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Factions are organizations or groups. In Morrowind there are 26 factions, most of which you can join. They are the:
Ashlanders, Blades, Camonna Tong, Census and Excise, Clan Aundae, Clan Berne, Clan Quarra, Dark Brotherhood, East Empire Company, Fighters Guild, Hands of Almalexia, Hlaalu, Imperial Cult, Imperial Legion, Mages Guild, Morag Tong, Nerevarine, Redoran, Royal Guard, Sixth House, Skaal, Talos Cult, Telvanni, Temple, Thieves Guild, and Twin Lamps.
The only dealings you'll have with the Census and Excise Faction is with the 2 members in their offices in Seyda Neen and 1 guy in the St. Olms Waistworks when you are information gathering for Caius. The only Dark Brotherhood encounters you'll have will be with the assassins from and in Mournhold. The only Hands of Almalexia members are the 5 guards in the High Chapel in the Mournhold Temple. The Royal Guard are all the folks in the scarlet (pink?) armor in Mournhold. The Sixth House is your enemy – all the "dreamers" in the game, and theoretically, the Dagoths and all their minions, too. The Skaal is to Solstheim what the Ashlanders are to Vvardenfell – the natives. You're made an honorary Skaal, but you're never quite in the faction the way you will be with the Ashlanders, so for all intents and purposes, it's non-joinable. Lastly, the Talos Cult is limited to the 2 people (one less than the Census people, and that much more embarrassing) that are involved in an Imperial Legion quest. In fact, after the Imperial Legion quest, there is no more Talos Cult, if you get my drift.
You can join the Ashlanders, Blades, Clan Aundae, Clan Berne, Clan Quarra, East Empire Company, Fighters Guild, Hlaalu, Imperial Cult, Imperial Legion, Mages Guild, Morag Tong, Nerevarine, Redoran, Telvanni, Temple, Thieves Guild, and Twin Lamps.
As you perform the main quest, you will have to join the Blades, you will end up joining the Ashlanders, and by default, you are the Nerevarine (a "faction" of one (you)). If you free enough slaves, you are automatically considered a member of the Twin Lamps.
The East empire company may only be joined in Bloodmoon.
You may join as many of the other joinable factions as you wish, as permitted based on your qualifications.
Each faction has favored attributes and skills that you will need to have good scores in to join. You will need to improve in these scores and complete quests to increase in rank within the faction. I will try to remind you to check in for advancement in some faction walkthroughs, and tell you where you should be in their rankings, but not always. Of the joinable factions, most allow you to assume the highest title within that faction if you do enough quests and are representative enough of that faction's ideals. You will not be able to progress beyond a certain rank in the Ashlanders or Blades.
Different factions get along with each other in different ways – some are at odds while others are allies. These faction-relations will affect your dealings with people based on which factions you belong to and which factions they belong to. These faction-relation modifiers affect disposition. In each faction walkthrough I have listed the factions and their relations with each other as friends or foes. Within each list I have grouped the factions from most hated to least hated and most friendly to least friendly, grouping factions that are held in similar regard within semicolons. All factions are maximally friendly with their own members.
Some groups of factions deserve more details.
The Great Houses
There are three "Great Houses" in Morrowind that you can join: Hlaalu, Redoran, and Telvanni. You may join one of the Great Houses, but that membership will preclude you from joining either of the other two. The Houses do not get along with each other, so joining one affects your relationship with the other two, as well as various other factions in Morrowind. All of the Houses will provide you with the opportunity to build a stronghold of your own once you are of appropriate rank. Each House will also require you to find a sponsor within the House, gain the support of the other Councilors in the House, and eliminate an NPC whose progress has rivaled your own, in each of the other two houses.
There are three major guilds in Morrowind that you can join: the Fighters Guild, the Mages Guild, and the Thieves Guild. Unlike the Great Houses, you can join all three and even advance to the highest rank in all three. You don't build strongholds and you don't need sponsors in the guilds.
Like the Great Houses, they have their share of intrigue and have various friends and foes among the other Morrowind factions. The real "buyer beware" here will be in regards to the Fighters and Thieves – their respective quests will force you to choose sides and make some moral decisions about where you stand allegiance-wise as you play through.
If you are going to do both the Fighter and Thieves quests, with the intention of becoming the head of both guilds, I would recommend doing (or completing) the Thieves Guild quests first. The Mages Guild has no conflicts with the other two guilds.
The Mages and Fighters Guilds have actual guildhalls in major cities, and they are thusly named. The Thieves Guild operates out of less apparent locations, frequently the local cornerclub.
The Vampire Clans
There are three Vampire Clans in Morrowind: Clan Aundae, Clan Berne, and Clan Quarra. Every vampire you meet belongs to one of those clans, and these clans are sworn enemies of each other. When you fight a vampire, every physical attack they make on you has a chance of infecting you with the disease, "porphyric hemophilia." You will get a text message when and if this happens. A resistance to common disease will reduce your chances of becoming infected. If you are infected, you can initially cure the disease with a cure common disease scroll, spell, or potion. However, if three full days have passed, and you rest, you will awaken as a vampire of the clan that infected you.
As a vampire, you enjoy some benefits and some penalties. Your strength, willpower, and speed abilities will go up by 20 points. Your acrobatic, athletic, destruction, hand-to-hand, illusion, mysticism, sneak, and unarmored skills will increase by 30 points. You will gain the spell, "Vampire Touch," that absorbs your opponent's health, and a Levitate spell that is unique to vampires (it's more powerful than the normal spell of the same name). You'll be immune to paralysis and common disease, and have a 50% resistance to normal weapons. Each clan also has it's own special abilities.
But, as a vampire, you will no longer regain lost health when you sleep. You'll be haunted by vivid dreams, and wake up just as weak as you were when you laid down to rest. The way you'll gain back lost health is through your Vampire Touch spell.
You'll take damage from the sun, so you're best off sticking to nighttime travel and staying inside during the day. And you're not going to be well liked. Many people will avoid you and refuse to interact with you, and some will outright attack you. This will impede your ability to perform quests, use public transportation, and progress in most quests. Other than your own clan members, members of the Mages Guild, and the Telvanni, you're not going to be tolerated much.
Once you have become a vampire, there is only one way to get cured. Once you are cured, you can never become a vampire again. Similarly, you can only join one vampire clan, and that'll be the clan you're in till you die or are cured.
Additionally, a vampire of any clan can perform 6-7 general vampire quests if you know where to go and how to activate them. These quests are listed separately from the Clan quests, and the quest to cure vampirism is included therein.
Fortunately, you can usually become a vampire, perform the vampire quests, and then go about getting cured without messing up your chances of completing the main quest or any other set of quests.