The most popular options are:
1.) Brookline: safe, expensive, homogeneous, and quiet. The Coolidge Corner and Washington Square neighborhoods are a little more lively. Expect to pay at least $1000 if you want your own place, or $650 if you are sharing. It's a 10-20 minute walk or 10 minute bus ride from school.
2.) Jamaica Plain: where all of the students from San Francisco prefer to live. It's lively, diverse, and quickly gentrifying. A few years ago it was largely Latino, but the people who thought Brookline was too chill or too expensive moved there and raised the prices. Expect to pay at least $800 for your own place or $500 to share. Much of it is safe but there are some parts to avoid. It's a 10 minute bus ride from school.
3.) Mission Hill: right next to school with really low rent, but not the safest part of Boston. Also, there's not much to do. Expect to pay $700 for your own place or $400 to share.
4.) Cambridge: many students live there because of the shuttle that runs between the Cambridge campus and the Longwood area. Rents are comparable to Brookline and Jamaica Plain and most parts are quite safe. The shuttle ride at peak hours can take 30 minutes and it runs every 15 miuntes.
5.) Back Bay: ultra chic (bring your black leather and BMW) and expensive. Expect to pay $1200 for your own place or $900 to share. It's along the route of that same shuttle I described earlier.
6.) Brighton/Cleveland Circle: it's where Boston College is located so it has a very homogeneous collegiate atmosphere. The rent is fair and the commute is 30 minutes by train.
Plenty of people live further out but I would avoid it first year if I were you. I hope this helps and let me know if you have any more questions.