A Roman apartment building was called an insula (insulae in the plural). This word means 'island'. Think of it as a metaphor for how the insulae rise up from a 'sea of streets' to provide habitation for the lower and middle classes.
Insulae were owned, in full or in part, by wealthy individuals and were usually seen as investment properties. One would not 'purchase' a unit within an insula any more than one would 'purchase' a room within a hotel. If your family wasn't rich enough to own a domus (a house), you would have to make do with an apartment in an insula. To give an idea of the rarity of home ownership: in the 4^th Century there were roughly 44,000 insulae in Rome's city limits but only 1,700 domus.
Units on the ground floor, called tabernae (the origin of the word 'tavern'), would open onto the street and would be rented by merchants. Such market stalls can be seen in Pompeii. These could be anything: a lawyer's office, a restaurant, a jeweller, a liquor store.
If your building was fancy it would have a public latrine attached to it. This would be used by residents on upper floors, perhaps even by residents of neighbouring buildings. A public latrine would have been a long bench with many holes in it. If you looked through the holes you'd see a gutter with a constant stream of water rushing along it, carrying waste away. If you looked around you you'd see a large crowd of men and women chatting: latrines were a common social gathering place. There would also be sticks next to each seat. These would look something like the toilet brushes that we use to clean our toilets today. You would use this brush to scrub yourself when you were finished. If someone put the stick back into its holder facing the wrong way, you'd get shit on your hand when you grabbed it. This is supposedly the origin of the phrase "getting the wrong end of the stick" (and its variants).
The lower floors of the insula would feature central heating, indoor plumbing, sound walls, and spacious accommodation. The higher the floor, the fewer the amenities, the thinner the walls, and the smaller the rooms. Unlike today, it was desirable to live as close to street level as possible: living on the top floor of an insula meant many flights of stairs, cold winters, hot summers, and a complete reliance on restaurants and public latrines for basic human needs.
Owners of insulae often had callous disregard for their more impoverished occupants; shoddy construction was the norm on higher floors and fatal collapses were commonplace. Augustus restricted the height of insulae to ~20.7 metres; Nero later lowered this height to ~17.75 metres. Insulae were never more than nine storeys high. The Insula Felicles (the "Happy Isles Apartments") is an example of one of the tallest. A hand points to it in this close-up of a model of Rome.
Here are some images from Pompeii to show how the walls of some of these units were decorated. Note that in this particular case (Insula 9) only the lowest level of the building has been preserved. This means that the units represented on this website were the largest and most richly appointed.
Edit: On the subject of the Insula Felicles, one of the tallest insulae in Rome: Tertullian, a Christian writer, had some rather nasty things to say about the pagans in "Against the Valentinians". He uses the Insula Felicles in a metaphor, which I'll quote from this source.
Ennius, the Roman poet, was the first to mention (with a straightforward meaning) "the great halls of heaven," because of its lofty position or because he had read in Homer of Jupiter's feasting there. Now as for the heretics--it is a marvel how many pinnacles on pinnacles and towers on towers they hang, add, develop on the house of each god of theirs. Well, perhaps even for our creator these Ennian halls have been distributed like apartments. Perhaps they have various shops built on in front and assigned to each god by floors--as many floors as there are heresies. In this way the world becomes an apartment house; indeed, you might think the celestial flats are the Happy Isles Apartments, located somewhere. There even the Valentinian god lives--in the penthouse.*
*Recalling, of course, what it means to live 'in the penthouse'. ;)
/u/kwonza asked a follow-up question that I answered here.
/u/theGentlemanInWhite asked another follow-up question that I answered here
There are many follow-up questions below this answer, read them all!
Gold? In my insula? Thank you, stranger! As a bonus, have an accessible documentary on Roman architecture.
If you don't mind an additional question (if you do I'll remove this myself), how large were said apartments both in number of units and the square footage of said units?
As an addition - how was the paperwork organized. Say I'm a wealty merchant who sold his lands in Spain and decided to move to Rome for a retirement. Where do I go with my cash?
