Apartment building 

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Flats / Apartments: A multi-unit dwelling made up of several (generally four or more) apartments. 
Flat: An apartment, especially one taking up an entire floor of a house with several flats.
Apartment: A relatively self-contained housing unit in a building which is often rented out to a family or one or more people for their exclusive use. Sometimes called a flat.
Studio Apartment: A self-contained unit with one main room, one bathroom, and some closet space. There is no distinct bedroom in a studio: sleeping, cooking, dining, living is all done in the main room
Duplex: Two separate residences, usually side-by-side, but sometimes on two different floors. The former often looks like two houses put together, sharing a wall (see semi-detached); the latter usually appears as a townhouse, but with two different entrances.
Penthouse: The top floor of multi-story building
Apartment tower, block of flats or tower block: a high-rise apartment building
Condominium: A form of ownership of an individual apartment and a percentage of common areas
Co-op: A form of ownership in which a corporation owns the entire apartment building or development and residents own shares in the corporation that correspond to their apartment and a percentage of common areas.
Garden Apartment: A building style usually characterized by two story, semi-detached buildings, each floor being a separate apartment.
Housing project: government-owned housing
Tenement : A multi-unit dwelling made up of several (generally four or more) apartments (i.e. an apartment building). In the United States the connotation implies a run-down or poorly-cared-for building.
Four-plus-one: An apartment building that has four floors of apartments on top of parking. It was particularly popular in Chicago during the 1960s and 1970s, especially on the city's north side.
Triple decker: A three-family apartment house, usually of frame construction, in which all three apartment units are stacked on top of one another.