What should I see in Boston in two days' time?

Boston is a very walkable city. A good way to get to the must-see places is to follow  The Freedom Trail

Mains spots are  Public Garden,  Quincy Market,  Faneuil Hall Marketplace,  Park Street Church or the  Boston Public Library,  Old State House.

Beacon Hill is to me the prettiest neighborhood in the city.

Also a good way to start your day could be running at the  Charles River Path or close to the the  New England Aquarium Whale Watching.

A bit out of downtown the JFK Library is an interesting museum. You can also check  Harvard Square.

Hope that helped!

Boston is indeed a VERY walkable city and depending on what you enjoy, there's something for everyone. Clement gives a great list to start so I'll add to it. If you enjoy art I highly recommend checking out the  Museum of Fine Arts and the  Institute of Contemporary Art. these are both great spots for art and each has it's own flavor. If it's somewhat warm out and there's no snow on the ground definitely check out  Boston Common and  Public Garden.  Faneuil Hall Marketplace has some decent shopping but if you're looking to really do some shopping, hit up Newbury st in Boston's Back Bay neighborhood. While there, walk 2 blocks over and take a stroll up Commonwealth Ave and marvel at all of the amazing brick rowhouses.

No visit to Boston is complete without a stroll through the  North End Boston's "Little Italy." Make sure to pick up a box of cannolis from  Mike's Pastry, if you enjoy cigars make a stop at  Stanza dei Sigari it's a cool basement "speakeasy" type cigar lounge with a full liquor license. While there's no such thing as a bad restaurant in the neighborhood, my top picks are  Neptune Oyster which (in my opinion) has Boston's BEST oysters, BUT, it can be very tough to get a table and they don't take reservation. My other two picks for the North End are  Antico Forno and  Strega Ristorante

Other good food spots are  Toro but don't get there any later than 5pm or else you'll be waiting for hours to get a table, but it's well worth it! Best tapas around!  Picco in the South End is a great pizza restaurant. I would also advise checking out  Lucky's Lounge for excellent food AND drinks near the waterfront on Congress st. 

I could go on, but I don't want to overwhelm!

Follow the Freedom Trail, it's painted on the ground and takes you on a great walking tour. Don't forget about Fenway Park and maybe Harvard.

I agree with TJ on Fenway Park. They give tours that take you around historical aspects of the park (including swag from when the Mets beat them in the '86 World Series -- a personal favorite moment of mine!). You also get to sit at the top of the green monster. It's a great tour if you're a sports fan.

After you've seen all of Boston proper, ride a bike over the water to Harvard Square where there are some cool historical attractions and an amazing farm-to-table restaurant, Henrietta's Table.

Have fun!

Whenever I have friends in town that haven't been to Boston, we always end up at Top of the Hub in the Prudential Center for drinks. It's an expensive, and some may say overrated restaurant, but they have great drinks and amazing views of the city. If you go in the evening, you can often catch a small jazz group. 

If you're looking for a great meal, I suggest Eastern Standard which is right off the Kenmore T stop and around the corner from Fenway. It's a bit on the pricey side but has fantastic food and an amazing bar. If you're looking for a less expensive bite from there, check out their bar area at late night. They start serving discounted items from the kitchen around 11 every night. They write up each unique menu item on the mirror wall behind the bar and every night it's different (it's based on what the kitchen has leftover after dinner service). Each item ranges from about $3-$7. 

And as everyone else says, Boston is super walkable and it's the easiest way to get around! I suggest taking a walk along the Charles, it's quite beautiful. 

go on The Freedom Trail very historic and lovely for food

Legal Sea Foods is a must also

go to Faneuil Hall Marketplace

have lunch at Durgin-Park

have fun

boston is a great city