Q: How does the Apartment System work?
The Sims are designed so they don't have a sense of personal ownership: every object on a lot 'belongs' to every Sim who lives there, no matter what 'room' it is placed in. (Sims will develop 'relationships' to objects they use frequently, but that doesn't prevent another Sim from using one Sim's favorite chair, for instance). The Sim LogicalApartment System allow you to assign particular Sims to a room token(s), which restricts the use of apartment hacked objects in the room(s) to just the assigned Sims.
The system works by using tokens which are placed in individual rooms and assigned an 'apartment number'. Every token on a lot that has the same 'apartment number', and will belong to the any Sims who are assigned to that 'apartment number'. The tokens work with objects specially hacked for the apartment system: any apartment hacked object that is in a room that has an apartment token in it will only be available to Sims who have been assigned to the token.
Q: How do I set up the apartment tokens?
This tutorial with pictures describes how to set up your apartment Tokens.
Q: Why can't I just use a locked (or apartment) door to keep apartment dwelling Sims from using their neighbors' stuff?
Sims are aware of every object on the lot they are on: the objects literally 'advertise' themselves to the Sims, telling them, 'use me, I'm fun!' or 'use me, I'll take care of your bladder needs'. Sims are constantly being bombarded by messages from the objects around them, and responding to them based on their needs bars (so a Sim with a red bladder bar will hear a toilet advertising and use a toilet. Not necessarily the toilet in the bathroom right next to him, but often the toilet on a different floor) and also based on their personality (some objects are more attractive to 'playful' Sims for instance). What the apartment system does is allow you to assign particular Sims to a room token(s), which restricts the use of apartment hacked objects in the room(s) to just the assigned Sims.
Most importantly, it stops the objects from 'advertising' to any except the assigned Sims. Before the apartment system was developed, many people attempted to create apartment situations by using 'lockable' doors which only allowed access to particular Sims. While this was helpful, apartment living with Sims was still very difficult because Sims would continue to try to use the objects in other apartments, even though the door would not let them through. The player would have to be very vigilant, making sure apartment dwelling Sims wouldn't die of starvation wandering the halls with an uncooked meal they were trying to prepare on their neighbor's stove, even though they have a perfectly good stove in their own apt. Sims will unfortunately 'obsess' over an object they are trying to use and will not give up until you 'X' the action out of their queue. The apartment tokens only work with objects specially hacked to work with them, any Maxis objects or custom objects not specifically hacked for the Sim Logical Apt. System will still advertise to all Sims on the lot.
Q: Can I use non-Apartment Hacked object in my apartments with the room tokens?
You can but only objects hacked for Apartments will respond to the tokens, so any objects you place in an apartment that aren't apartment hacked will be available to all the Sims on the lot. Using non-apartment hacked objects won't interfere with the apartment hacked objects, but the more attractive the objects are to the other Sims, the more time you will need to spend keeping Sims out of each other's apartments. Things like decor and lighting, or any objects Sims don't use on their own are completely 'safe'.
Q: Can I use apartment hacked objects in non-apartment lots?
Yes, you can. The Apartment hacked objects function like 'normal' objects when you place them on lots or in rooms that have no apartment tokens in them, so you can use them like normal objects on regular lots. You must have the apartment controller and tokens installed in your game to use these objects safely (to avoid crashes), but you don't actually have to have the controller and token on the lot. (they have to be on the lot to work as 'apt' objects though).
Note: apartment hacked objects required a bit more 'processing time' than normal objects, so using apartment hacked versions in place of regular objects could *possibly* make your game run slower. I use apartment objects on 'regular; lots all the time and notice no difference. This will probably only effect very slow computers.
Q: Can I use the apartment system to create private roooms on single family lots?
Though the system was created to help players set up apartment buildings, the tokens and controller can be used in many other situations to create 'private spaces' for your Sims. I use them in single family homes to restrict bed/bathroom suites, so Sims don't tramp through other Sims bedrooms to use a toilet, when there are other bathrooms in the house for everyone to use. You can give the kids a study area in their bedroom (apt hacked computer and bookcase) and Mom won't decide to read her book sitting in the kid's room. Kids can have private bathrooms so adults don't barge in while they are taking a bath (guests are so rude! lol).
The system only works on objects that are apartment hacked, so you have maximum flexibility while using it. I have a single family house where the daughter has a bedroom with a private bath. The bathroom has all apartment hacked plumbing, which keeps her step father and the uncle that visits frequently out of there. (really, it was very annoying! There's a bathroom right outside the kid's bedroom for people to use! But no, they have to use the kid's toilet, even while she's having a bath! lol) I use an apartment hacked door on her bathroom, which also keeps the pets out, so the girl is never trapped in her tub by a dog falling asleep next it while she's taking a bath. (this has happened to me!).
The girl's bedroom also has an apartment token, and she has a computer and bookcase only she can use. But her bedroom door is a normal door and her parents and pets can come in at any time to talk with her or play dollhouse with her. (well, the pets can't do that...). In the same house, I have the master bed and bathroom set up as an 'apt' for the parents. The rest of the house is unrestricted, there are no tokens in any other rooms, just like any other house.
Q: I assigned Sims to their apartments, but they haven't moved in!
The apartment system allows you to 'give a share' of an apartment to any Sim that exists for that neighborhood, including the Townies, vacation tourists, Somebodies and Anybodies, all the Sims in your neighborhood, pets and strays (depending on which EPs you have installed). It doesn't move them onto the lot however. You can use the 'move in plant' from Sim Logical to move Sims into your Apt. NOTE: because you can assign a 'share' to non-resident Sims, that means you can give a share to any Sim that you have as a frequent visitor.
More sites with Apartment hacked items and information:
Home of the Apartment hack, this site has the core downloads required for this system along wiht may Maxis clones that have been apartment hacked (and a few unique goodies). ake sure to get the token and controller far and get the apartment hacked meals that aptarment hacked fridges require. A backup site for these downloads was set up as a yahoo group during Simlogical's "downtime" Simlogical Apartment Archive Group 
Madoria offers quite a lot of her sets in an apartment hacked version but you'll also find a few sets at this site that are only available as Apartment Hacked items.
Many unique residential sets that are only available as apartment hacked items. 
A sister group to Shiny Things, Shiny Things shared offers sets of apartment hacked objects for more modern time periods as well as some essentials for earlier eras. 
This apartment hacked extention of Alex Pilgrim's normal sites offers both lots and objects, including several tiled bathrooms and apartment hacked shop items to made to allow you to simulate shops on residential lot.