Most Rome guidebooks advise to take the non-stop, "express" train from Fiumicino airport to Rome's Termini station.  This train is known as the "Leonardo Express."  This is a good idea if your hotel is: 1) near the 'Termini' train station, or 2) on the red Metro line (Line "A") that goes toward the Spanish Steps, the Vatican, etc.  This train only leaves every 30 minutes, but tends to be very crowded, and can thus be a bit stressful.  A one way ticket costs 11 euro (as of Sept. 2007).

If your hotel is along the blue Metro Line (Line "B") that goes to the Colosseo, the EUR., etc, then you might instead consider taking a regional, non-express train from FIumicino airport to the Roma Ostiense station (convenient also to connect with the regional trains to Ostia) or Roma Tiburtina station (also convenient for example if you need to take a bus to a nearby town, as the bus station is just outside this train station).  Regional trains from the airport come more frequently (every 15 minutes) and, although they stop at all stations,  they are not much slower than the "express" train -- for example, the 'Leonardo Express' takes 31 minutes to Termini station, whereas the regional train takes 30 minutes to Ostiense station, 45 minutes to the Tiburtina station.   A one way ticket on these trains costs 5.50 euro (as of Sept. 2007).

Find out the schedules for both lines on the Trenitalia website -- http://www.trenitalia.com/en/index.html (enter 'Fiumicino Aeroporto' as point of origin).