I've been away from Boston for a couple years, so things might have changed a bit, but here's what they were like while I was there. Keep in mind these are gross generalizations, but can give you a taste for the areas:
North End - oldest area of the city, apartments tend to be very small, great "village" feel to the neighborhood (markets, small shops, etc.), and I always saw a lot of kids playing in the small parks, which led me to believe it would be a good family area
Beacon Hill - for the most part where all the lawyers, etc. that work downtown live. Very nice, very expensive, and (for the apartments I saw) very small. A pretty tame area. Quiet, and not a whole lot of excitement. Right next to Boston Common.
Back Bay - Wide gridded boulevards, Newbury street has lots of high end shopping, apartments tend to be bigger, expensive, gets crazy during the weekend from shoppers, many bars that tend to attract the college crowd, nice brownstones
South End - similar to the Back Bay but without the whole "fake" feel, large gay population, nice big apartments, good eateries
Fenway - close to Fenway Park, can be "transitional" in certain areas but "hip" in others, apartments are bigger, parking is horrendous during Sox games, newest part of Boston proper. I lived in this area for a while and loved it. I had a huge apartment for cheap (by Boston standards) that overlooked a park and I was very close to the esplanade for some decent in-city biking.
Brookline - technically a suburb, but (like much of metro Boston) you would never know it. Tends to be upscale, family-oriented. Tends to be more low key and quiet. A generally pleasant area with a mix of housing styles.
Brighton - next to Brookline, VERY busy, VERY college-oriented (read: cheap rent), lots of hole-in-the-wall bars/restaurants, big mix of housing from triple deckers to brownstones to large victorian houses.
Cambridge - my favorite area, just over the Charles, vastly different areas from South to North, big mix of housing styles, lots of things to do, just plain great
Somerville - never spent a lot of time there, but I understand some great deals can be had for housing, some areas can be relatively rough, areas close to Cambridge tend to be better
If you're looking at $2K a month, you can live in any area. Obviously you'll get more for your money depending on location, but I don't think you have a thing to worry about. Everyone I knew there tended to move a lot depending on what neighborhood vibe (or roommate situation) they wanted. Have fun!
