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.
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. Rent costs more and heating costs less after each upgrade, but since the heating can be turned off to save money and the rent cannot, upgrading leads to a monetary net loss, unless one upgrades to a class 7 dwelling on day 12 and downgrades the following day, after the inspector's new-found wealth is confiscated, allowing the inspector to retain 75 credits.
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. The only option for downgrading is moving back to a class 8 apartment. The refund amount depends on the class of the apartment: 75 credits for downgrading from class 7, 150 credits for class 6, or 225 credits for class 5.