An apartment is where the inspector and his family live while in East Grestin. Their initial apartment is provided to them when the inspector is appointed after the October 1982 Labor Lottery. Their apartment building (or one similar to theirs) is briefly shown in the intro sequence but their own apartment is never seen.
Gameplay
The apartment affects gameplay at the end of each day as daily rent and heating expenses depend on the class of the apartment. A better apartment costs more without providing any benefits so upgrading only makes sense when trying to unlock the Worker's Bestachievement.
An option to move to a better and more expensive apartment becomes available occasionally (see forum for unconfirmed theories). Moving to a new apartment costs 100 credits. Downgrading to a lower-class apartment refunds 75 credits and will be available on the day after the move-in.
Available apartments
Class-8 - This is the initial apartment provided to the inspector and his family at the beginning of the game.
rent: 20 / 25 credits / day (rent increases by 5 credits on day 4)
heat: 10 credits / day
total: 30/35 credits / day
Class-7 - Available at the end of day 12 at the earliest.
rent: 30 credits / day
heat: 10 credits / day
total: 40 credits / day
Class-6 - Available at the end of day 17 at the earliest, if already living in a class-7 apartment.
rent: 40 credits / day
heat: 5 credits / day
total: 45 credits / day
Class-5 - Available at the end of day 22 at the earliest, if already living in a class-6 apartment. Class 5 is "the best a worker can have" and moving in unlocks the Worker's Best achievement.