The type of living arrangements that Romans had depended on what they could afford, much like today! The poorer classes often lived in apartment buildings, called Insulae. The upper classes lived in townhouses, called a Domus, and often they would have a Villa out of town as well.
Insulae were developed in the late first century BC, due to population pressures in many cities. Before their development, insula instead refered to rectangular building plots in towns. They then became the most common form of housing, and occupants rented the apartments from landlords.
Insulae were often several storeys, with six to eight apartments all up, and the street level often being occupied by shops. Usually, the high up the apartments were, the poorer the living conditions were. Entire families might all be crowded into each apartment in an insulae. The higher storeys lacked running water, and so had to get water from public fountains. They also had to use public latrines, as they did not have toilets of their own.
Because in earlier times they were often made from wood and mudbrick, there were several hazards to living in an insula. Fire was often a big risk, and insulae were also prone to collapse. In later times, they were made from fired brick and concrete, which made them more safe, however improvements in sanitation did not seem to follow. Regulations were imposed on them, Augustus limitting their height to five storeys, and Nero added fire regulations.
This was the town house of the wealthy of ancient Rome. Unlike the Insulae, a domus only housed a single family.
The design of the house depended on climate, conditions of the particular city, when it was built, and the wealth and taste of the owner. The most popular style of domus was the Atrium-Peristyle house, also known as the Pompeiian House, after where it originated. The rooms of the house were positioned around the atrium (a large, open hall) and the peristyle (a sort of indoor garden). There were rooms for shops next to the entrance of the house, which led into the atrium. Between the atrium and the peristyle was the Tablinium, where the master of the house conducted most of his business. Family records were kept here. The room was informally separated by hangings. Other rooms might include bedrooms (cubiculum), kitchen (cucina), dining rooms (triclinium), reception rooms (oeci), and lavitory. Wealthier Romans might also have private baths and a library. For more info on each room, walk around the Villa of Thea Didius, keeping in mind that whilst it is as historically accurate as possible, it is written in role-play.
Another style was the strip house, in which a long side faced the street and wings extended behind.