I presume you are talking about the high speed trains (Frecciarossa, Frecciargento, Frecciabianca).
It depends on the fare you purchase:
1. Super-Economy fare: These offer the highest discounts, with tickets as low as 9 euro. These fares start selling 120 days prior to travel day and disappear quickly since only few seats are sold at this fare. Once you book a train at this fare, you can't change it or get a refund if you don't take the specific train you book. You are out of luck and you lose your money.
2. Economy fare: These offer a lower discount, but they also disappear quickly as only a limited number of seats per train are sold at this fare. The travel time/train can be changed but you have to pay the difference with the full Base price and a fee. You must do so prior to the departure time. No refund allowed, you can only change the time of departure on a different train or even a different itinerary, but can't get your money back if you want to cancel your trip totally.
3. Base fare: This is the full fare. If you pre-purchase at this fare, you can change the reservation or even get a full refund if you need. You can buy at this fare also at the station, the same day or the day before. It's very rare that a train sells out all seats, so the Base fare is always available. At most they run out of the 2nd class (aka 'standard' class in the Frecciarossa) on some popular routes in the morning (like Rome to Florence), in which case you might have to buy a higher class or wait for the next train 1 hour later. Later in the morning and in the afternoon they never sell out any class.
http://www.trenitalia.com/tcom-en/Offers/Our--prices/Our-prices
If you take a Regional train, then there are no discounts and there is no reason whatsoever why you should buy them ahead of time from the US. Just buy them there. They are like bus tickets. Just stamp them before boarding and they are valid for 4 hours till your destination. No seat reservations, if it's full you just stand up (like a bus or the NY subway).
So tell us where you are going (from where to where and when exactly) and we can probably give you better advice.