There are three types of public transportation in Rome; Metro, Tram, and Bus. All three use the same tickets. You can purchase individual tickets at Tabacchi (tobacco / candy shops located all around Rome) or at Metro stations.
You can purchase unlimited one-day, three-day, one week and one month tickets at Metro stations in person or through a machine. For now, I’ll just talk about the one-ride tickets available at all these locations.
A single ride ticket gives you 100 minutes of unlimited riding on any city bus, tram (street car) or Metro (underground subway). The 100 minutes does not commence until you validate it on your first ride, so, the ticket can be in your wallet for days or weeks ready for use. If you do plan to use mass transit, I recommend you purchase a few tickets and have them in your pocket. It is sometimes hard to find a place to purchase tickets, especially late at night.
You enter the Metro by inserting your ticket into the slot on the turnstile. Thus validating it and beginning your 100 minutes of use. It pops out of the top of the turnstile, you should pocket it to prove you’ve paid, and to have in case you transfer to a bus or tram and want to prove you have a validated ticket,
You can get on busses and street cars through any door. If the ticket you hold has not been validated, insert it into the orange machine bolted to one of the handrails. It will stamp a validation and return it to you. Pocket it. for the next 100 minutes you can ride any bus or tram without pulling it out, but it is there incase a transit inspector asks you to produce it. You only have to process it again if you enter the subway using a pre-validated ticket. 
At the end of the 100 minutes, the ticket is worthless. You might as well throw it away, or save it for your travel scrap book.