If you are a tourist to Boston,, you'll probably stick close to downtown and the Freedom Trail and the waterfront. There's nothing wrong with that, but any Bostonian will tell you there is more than that. Much more. Boston is known as a city of neighborhoods and even these main divisions outside of downtown are subdivided into smaller, navigatable parcels. Boston is a livable, walkable city well served by public transportation. If you don't go out to the outer edges to explore, you'll be missing 80% of the Boston experience, at least.
Few observers agree as to the main divisions, but here is a sample:
Beyond downtown Boston and the Back Bay, the South End, the North End and the West End, you can think of these neighborhoods as starting points when you make your mental map of Bean Town:
Proceeding clockwise: Charlestown. East Boston (Eastie). South Boston (Southie). Dorchester (Dot). Roxbury. Hyde Park. West Roxbury (Westie). Allston/Brighton.
You'll notice that some places have been left off this list that probably belong as stand-alone communities. Mattapan, Jamaica Plain, and Roslindale for instance. The above list takes into account only the largest divisions. There are many, many sub-divisions. Boston is a place best experienced, whether you live in a niche of it or explore a corner on foot. Either way, there is no real best neighborhood. They are all good in their own way. Boston is good overall and worth exploring and learning. There is no other city like it.