Boston is one of my all time favourite American cities. It has a very different feel compared to the majority of the state capitals as it's very heavily influenced by the immigrants as it was one of the main ports hence as Fox has mentioned it has a lot of very historic buildings, and doesn't follow the standard grid layout.
It has an exceptional public transport network, and I wouldn't recommend driving as it's one of the harder to navigate by road and parking is expensive. A charlie card can be bought for $15 (may have been cheaper) which will give you unlimited usage of the subway AND bus network. This can also be used on the "tram bus" link from the airport. The subways are all linked through the various lines so it's very quick to get to where you're going, although they can be somewhat crowded at rush hour!
There is a wealth of things to do, we had 6 days there in June/July last year and could have easily spent another fortnight there (our 3rd trip there as well!). You can walk the "Freedom Trail" which is a red brick trail you can walk and takes in a lot of the main historic attractions. There is also the obvious trip to the Boston Red Sox stadium at Fenway Park, multiple good shopping areas, and I would strongly recommend getting the U (Subway) out to Cambridge for a walk round the Harvard Campus and surrounding area. You can then walk back into the centre for a nice walk which will take you past MIT (the MIT museum is worth a visit, lots of cool robots and techie stuff). A trip out to the Sam Adams brewery is also good and they only ask for a donation and you get to try out lots of their beer in their tasting bar for free, easily reachable by subway.
I would also recommend taking a couple of days to visit Salem, you can either take an AMTRAK train for about $5 or there is a cracking Ferry which will take you/to from Boston. We got the train up and ferry back and wished we'd decided to stop a night there as there is lots to do! If you do go to Salem, the Witch City Segway Tour is a fun and slightly different way to get around the attractions!
There is a huge choice of food, obviously sea food is something that they have a reputation for and if you like crab/lobster etc you'll be in for a treat but with the immigrant heritage there is a huge choice of places to go and eat.
Back Bay is a nice area and close enough that it's a nice short walk into town or you could take the aforementioned subway. If you're going for more than a couple of days which I would recommend you do, then look into self catering as hotels in Boston are relatively expensive compared to some other cities and this can be a good way to do it particularly if you get somewhere near to a subway stop.
Boston is a very friendly city, we had people asking us if we needed directions, if you even look like you want to cross the street then cars will stop and wave you across (compared to other American cities where they positively try and run you down!). When we walked back from the Harvard campus we came across a fire station and were taking some photos (they keep the doors open) when two engines came back. The guys insisted we come in and get in the fire engine and take photos and let my girlfriend try the uniform on and get photos etc.
Oh and one last thing, the Duck tour is worth a go. It's an amphibious tour (one of the old world war 2 things) which takes you round the city then a splash in the bay. The tour guides are great and you'll see a lot of them driving around, punctuating the air with quacks and duck whistles.
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