I understand that it can be convenient (and preferable to driving) to take the train between Philadelphia and NYC and between NYC and Boston, but that one should make reservations. Is this Amtrak, or is there a regional transportation authority? Please forgive my ignorance, but I'm a Texas girl.

Yes, the train often beats driving especially city center to city center. I suggest you start with the Amtrak website at http://www.amtrak.com
You will need to make a reservation for the train you wish to take. There are two levels of trains running between these cities . The Acela service and the Regional service. The acela is a higher fare express service using newer dedicated train sets while the regional trains use standard coaches. There are generally two levels of service on each train, coach class and business class. The business class offers 2 + 1 seating whith some amenities, while the coach class offers standard 2 + 2 seating. There is usually a snack car offering quick sandwiches and drinks, but no dining car. Seats are reserved but not assigned, so on busy days you may have to search for an open seat.
The fares are not inexenpensive and are based on time of travel.
I prefer the train to driving and the trip between NYC and Boston follows the coast for a short distance in CT giving you a nice view of the beaches.
Enjoy your travels.

All the inter-city trains are run by Amtrak. Each city also has commuter trains, between the city and suburbs, that are run by local authorities.

The long distance trains are Amtrak - and theirs are faster and with comfier seats. It's possible to do the trip via a series of local commuter trains, but it would take forever (stopping at all the local stations) - and the seats are more like what you find on your local bus. Also - no refreshments. (Amtrak doesn;t serve meals, but does have a cafe car with snacks and prepackages sandwiches etc).

Do not bother with the Acela. It costs more and SHOULD be faster - but often really isn;t since the tracks are in such poor shape they don't allow for high speeds.

It's preferable to driving if you have to pay to rent a car, with substantial drop-off charge for a different city - and have to fight traffic.

I live in NYC and will drive to Boston (which is faster than train or plane) but only because of the side of the city I live on and have my own car.

Train won;t cost less than plane - but takes no longer (since you going city center to city center and don;t have to deal with trekking out to airports and long security lines) and is much more comfortable.

The track in the Northeast is OK. In fact the trains do reach speeds of 120 mph in parts, certainly not as fast as the TGV In Europe, but fast for American standards.In fact, the avg speed in the Northeast is over 89mph True, for a short hop between Philly and NYC, the Acela would save you 15 minutes but between NY and Boston it can save you an hour.