Where does one local accent stop and another start? Is there a gradual transition from one accent to another, or is there an abrubt change at some conceptual border between two areas?
There are actually linguistic maps of different dialects. What basically happens is that linguists determine the epicenter of a particular dialect (for example, Boston’s characteristic r-less dialect) and then draw concentric circles out from this epicenter. Kind of like ripples in a pond. The further away, the less it shows up.
Of course, these centers overlap. So a particular area can incorporate elements of both A and B. Depending on where they are, they might have more of A than B. Generally speaking, it is a fairly gradual transition
It’s impossible to draw definite borders simply because of the nature of language.
Well, a NY accent never becomes a Boston accent. It becomes a Connecticut/Springfield nonaccent, then becomes either a Worcester or Rhode Island accent (and within RI, it can be either a Pawtucket, Cranston, or Providence accent), and finally becomes a Boston accent when you enter the 128 beltway around the city. Of course, even when you’re inside 128, there’s a Revere accent, and many others, as well.
In short, what Andygirl said.
When I took linguistics in college, I was told that the dividing line is the Connecticut river.
The explanation give for this was that the people who settled eastern New England moved west, and for the most part stopped at the Connecticut river. (The Connecticut river forms the border between Vermont and New Hampshire; runs through Massachussets past Springfield, then runs through Connecticut through Hartford, and empties into Long Island Sound.)
I agree with the posters who indicated that linguistic differences can change gradually. But they also can change dramatically in a short distance.
So, while there may be no absolutely clear dividing line between region accents associated with Boston and those associated with New York, you might notice a reasonably significant change when you cross the Connecticut, especially in small towns.
I lived from 25 years in Boston, Providence, and places in-between. I can confirm that even within that short distance, accents vary considerably. A Quincy accent is different from a Somerville accent is different from a Dorchester accent.
fatherjohn- I live next to the CT river at school. 
Anyway, you are right about geography being a major and abrupt divider of accents. Mountain ranges are a classic example of this.
It should be noted that modern times (exposure to mass media in particular) have led to more standardization across the country, generally making many accents less obvious.
Another thing to keep in mind is that the perception of accents is often most distinctly held by natives of an area. I can distinguish between people who live in different parts of Delaware. To someone from Ohio, we might sound entirely the same.
For many words, American English sounds about the same for everyone. It’s in the vowels that differences tend to show, and you can have an amusing time of discussing accents by asking all of your friends to pronounce words like about, water, naked, caught, merry, and cot.
Also, people speak in different ways depending on the situation and who they are talking to. (This is called register.) In informal speech people speak less precisely, and it is there that you will most often hear things like dropping “-ing” in favor of “-in”.