If you're unmarried, you either have to live on campus or in BYU-approved housing.
Helaman Halls - the only normal set of dorms at BYU (since they tore down Deseret Towers). Mostly you share a room with one other person, and it comes with a sink. You can also get a shared suite, which means 2 shared rooms share a bathroom, but this is only available in one building. Women have the option of having a single room that shares a bathroom with one other room. A meal plan is required.
Heritage Halls - apartment-styled housing. Apartments usually have 6 people in there (3 shared rooms) but there are a few 2-person apartments (still shared though). A meal plan is no longer required - you can buy groceries from The Creamery and/or eat on campus.
Wyview Park - used to be married housing but became single freshman housing when they tore down Deseret Towers. Apartment-style housing, with 3-4 residents, with both shared and single room options. Rent includes everything but electricity which is divided and then posted to the student account. A meal plan is not required.
Foreign Language Student Residence - apartments with 6 people (3 shared rooms). You have to apply to live here, and be at an intermediate or advanced level of some language. The ones they offer are Arabic, Chinese, French, German, Italian, Japanese, Portuguese, Russian, and Spanish. You must speak the particular language while in your apartment, but there is a native speaker in each apartment to help everyone else do this. You have to have dinner with the other speakers of your living language Sun-Thurs.
Apartments on campus have cleaning checks every month. Your swimsuits have to be modest - no bikinis or speedos. You can't operate a business from your apartment/room.
For more specific information, the page for BYU on-campus housing is here.
BYU-approved housing means it's within 2 miles of BYU campus and also that you only live in houses/apartments with occupants of the same sex.
- the distance thing is controversial because some argue that the ones that are approved can then charge more than necessary and get away with it, and it also seems pretty unnecessary. If people want to commute from Orem or Pleasant Grove, that should be their own deal, right?
The Honor Code has restrictions on off-campus housing as well. Members of the opposite sex are never supposed to be in bedrooms and not even use the bathrooms unless it's an emergency. There is a curfew where people of the opposite sex are not supposed to be in the house/apartment anymore - it's midnight except for Friday nights when it's 1:30 am. They are also not supposed to be there earlier than 9:00 am. Because of this, you may see people hanging out on porches or walking around town or in parks or in cars after the curfew. It may make some people go to bed earlier but mostly it just moves the party outside. Enforcement of the curfew really depends from place to place - it's important to find out how your roommates feel about this, ideally before you even decide to move in because that could be a big issue.