The most popularly used word for 'car' in Italian is macchina (pronounced makk-eena). This word actually means 'machine' but, in context, will always be understood as 'car'.
Another possible translation of car is automobile (pronounced awto-mob-eelay) which breaks down into the prefix auto- (a derivative of the Greek word for 'self' or 'same') and mobile (which is the Italian equivalent of the English homograph 'mobile').
The word automobile is usually shortened to just auto although, confusingly, it is not the only mode of transport that employs that abbreviation - most urban public buses in Italy are referred to as autobus or auto for short.
Occasionally, you will hear Italians refer to their car as a vettura. This term actually originates from the word for a four-wheeled carriage, but is by no means as archaic as its origin may suggest.
A final term we can throw into the bag is the more colloquial mezzo. The word mezzo enjoys about two dozen different meanings in Italian, one of which is 'a means' as in 'a means of transport' or un mezzo di trasporto. This term, as its wording suggests, can refer to any form of transport, and travelling by mezzi will often mean by 'public transport' specifically. Nevertheless, it wouldn't be unusual for someone to refer to their vehicle as il mio mezzo or 'my means'.