The Leonardo Express is the name of the train that'll shuttle you from Rome's main international airport, Fiumicino, to central Rome's Termini Station. A Leonardo Express train leaves every half hour in each direction.

From Rome Termini Train Station:
Trains to the airport start at 5:52 in the morning, and depart any 30min, from the stations at Tuscolana, Tiburtina, Ostiense depart every 15min.
The last train of the day departs at 22.52 (10:52pm)

From Rome Fiumicino International Airport:
Trains to Termini Station start at 6:35 in the morning, and depart any 30min.
The last train of the day departs at 23.35 (11:37pm)

The ticket price is € 11.00.
The train covers the 37 km in around 31 minutes.