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.
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.