Its "communal entrance hallway" or "entrance of an apartment building".
Other synonym is "парадное".
One building can have multiple entrances which lead to a different parts of a building.
Мы живем в одном подъезде.
"We use common entrance to the apartment building".
Моя квартира -- в четвертом подъезде на третьем этаже.
"My apartments are in the 4th entrance, on the third floor"
Она работает консьержкой в нашем подъезде.
"She is a doorkeeper of the entrance we use"
Она убирает в нашем подъезде.
"She cleans the entrance (and staircase, and other commonly used indoor spaces - hallway, corridors, elevators) we use".
Last example illustrates the idea of подъезд as a "section of an apartment building that uses single entrance".
Driveway is also correct, but "entrance" is more commonly used in cities.