Taking the train is a nice, very pleasant alternative to flying to/from Boston  to New York City.  Particularly if you're traveling from city center to city center, it wins hands down on convenience.  For business travelers, people seem to have one preference or another but many swear they are so much more comfortable AND productive on the train.  For leisure travelers, it's nice to have the extra space and to enjoy the "countryside" scenery and coastal views no matter the New England season.

Advantages

Boston's airport is convenient to the downtown area. But all three of the major airports in NYC are a major schlep away from Manhattan via cab, subway (for JFK Airport), or slow, expensive bus. For the additional hour or two you'd spend in transit on the train, the convenience of the stations at both ends goes a long way toward making up for it.

If you can afford it, Amtrak's Acela Express would be the train to travel on. It competes with the commuter airlines so the prices are much higher than the regional trains but they are much nicer and faster.  Definitely the choice if you're traveling for business.  Either way the journey is quite beautiful hugging the coast for a good part of the journey.  Note: Once the Acela Express started service, all trains took the name.   But not all are "express".  Also note:  In practice, the Acela Express shaves about 45 minutes off of the Boston-New York run.

Logistics

The Acela high-speed train takes ~3.5 hours, whereas  the regular train typically takes ~4-5 hours.  (This would compare to about 3-4 hours by flight depending on the traffic getting to and from the airport.)

Trains connect Penn Station in New York (midtown Manhattan, West 34th Street) to both Back Bay Station and South Station in Boston.  Back Bay is a popular destination for many travelers whereas South Station is more "downtown" and connects to different subway (T) lines if you're continuing your journey.  See www.mbta.com for more information on local transport in Boston.

Prices vary (of course) depending on specials and how/when you purchase but generally run about $75-$100 depending on which train you take. (This would compare to airfare between NYC and Bos which can vary from roughly $100 to $600.)  The more expensive trains are faster and roomier, but the cheaper trains are ok.  As mentioned, Acela is the high-speed train -- as close to a bullet train as we'll see anytime soon in crowded New England with old (curvy) tracks. 

Booking

Amtrak accepts booking online at www.amtrak.com, with a major credit card. It's ever too "early" to book the reservation, but you have to pay when you book. If you want to be sure to get a seat, make the reservation as soon as you are ready. Most Amtrak tickets are refundable, as long as you aren't getting a special discount fare.

Tips & tricks: 

Booking early may save you money

You can get much better pricing for long distance trains if you book in advance.  As mentioned, you have to pay at time of booking or several days later. One approach is to go back on Amtrak's web site and just put in some random dates a couple weeks ahead. (Make sure they are the same day of the week and time period of your actual travel plans.) Then compare the pricing to what it would be when you actually want to ride the rails. If there are no real price differences, then book your trip several weeks out rather than months. There are more than a dozen trains a day that make this journey Monday thru Friday, slightly less on weekends, so a sell out is not that common. Also, this region is all about the business traveler so fares tend to be cheaper during non rush hour time periods.  There are also senior, student and AAA rates. 

A tip regarding luggage

It has been reported in the Boston forum that one can check bags the night before a trip to New York (better if before the last train down the night before) for free, and they will be there on arrival at Penn Station.  This is probably the case when you travel to Boston, as well.

If the Priority is getting to/from NYC and less on training...

You need to know there are options.  Driving is the option of last resort, having to swerve around the bombed out craters that line I-95 through coastal-industrial Connecticut, and the endless congestion and gran prix-style driving on the parkways closer to NYC.  Actually, the Chinese Buses and Greyhound would probably rank slightly lower for this writer.  Yes, AMTRAK ranks a solid 3rd best choice.  But the two best forms of travel would be by luxury coach.  Some recommend the Hampton Jitney or the Limo Liner as they are appointed and served like the airlines used to be (actual refreshments).  The former is definately upscale and the later feels more like being on a private jet without having to leave terra firma.  Great for the traveler who does his best to avoid solo car travel and wants the creature comforts one can take for granted in Europe and almost never find in the States.  With companies such as these two, it is possible to travel in the Northeast utilizing quasi-public transport and not feel like a refugee.