There are a total of ten factions that will allow you to join, but you can only join eight of them (you are only allowed to belong to one of the Great Houses). I do not include Vampires in this list of factions, even though you can belong to a vampire clan, because being a vampire means you belong to a dead-end faction. No one except other vampires and a few mages will talk to you until you are normal again. Sure, it's nice to have super-stats and skills, but it's awfully lonely and about the only thing you can do is get yourself cured.
Factions are your bread and butter. They provide services and quests for their members. Unless you really fancy just running around the countryside and sticking your nose into every hole in the ground that you find, your life will be much easier if you join one or more.
New players would be well-advised to join the Imperial Cult and run through their Lay Healer, Almoner and Shrine Sergeant quests in order to get a feel for factions and questing. Except for the Shrine Sergeant quests, they can be completed in a relatively short time period, without expending all of your funds on transportation and the like, and involve little to no heavy combat. They will provide you a good feel for how things work in the game and do it in a relatively safe environment.
Joining a faction is generally just a matter of asking. If the person you are talking to does not have the authority to enroll new members, they will tell you who does and where to find them. The major exception to this is the Morag Tong (the native Assassins Guild). Finding the one person who can make you a member of the Morag Tong is a quest in and of itself.
Most faction quests start off pretty simply -- retrieve a book, deliver an item, escort someone, etc. But they get more difficult and more dangerous as you move up through the ranks. And such danger and difficulty is the stuff of which heroes are made.
Getting promoted in any given faction can be a true pain. If you notice from the various faction pages, the attribute requirements are minimal. Most players will meet the minimum attribute scores for the first few ranks just by starting the game and the difference between your starting attributes and the scores required for the highest ranks is something like five points. It's the skill requirements that will bite you every time. Pay particular attention to the skills required by each of your factions and work on developing those skills. Unlike Daggerfall, magicially enhanced skills will not help you at all when it's time for a promotion.
As is true of real life, some factions simply do not get along well with others. If you happen to belong to two factions who despise each other, your life becomes a bit more complicated if you want both sides to keep liking you. Sort of like having two friends who detest each other and you're caught in the middle.
For the Great Houses, you will reach a point where you will need to find a patron before you can progress higher in the ranks. Your patron will always be one of the current councilors, though finding one to sponsor you can be a bit of a task, depending on the House.
For each of the factions, I have provided information about their requirements, rank structure, who they like/dislike, and who gives quests. The actual quest walkthrus are in the questing section. The exceptions to this are the Stronghold quests for the Great Houses. Since they all pretty much work the same, I have included them with the appropriate House.
If you break the rules and are expelled from a faction, you must seek out the proper person (usually one of the quest givers) to make amends. You are allowed to do this once. The second time you are expelled from the faction, it is permanent. The exception to this rule appears to be House Telvanni. If you are higher than rank 5 and lower than rank 10 and don't kill Aryon, you should be able to make amends an infinite number of times. I have not tested this, but that appears to be how the script runs. The Mages Guild appears to run similarly as long as you haven't killed Trebonius.