Hello Jill and Todd- I'm a Mass. licensed builder, and also a RE investor. I've been through this question many times.
Other than officially callling it an apartment, if the town doesn't allow it - via the zoning ordinance- there may be other options, like an in-law apartment or an "au-pair" suite.
Typical Mass. state building code is that the finished ceiling height must be at least 7'0, for a finished basement area. If you don't have that (after sheetrock and a finished floor), I'd give up. There are other building requirements, and it would be smart, and free, for you to have a local general contractor do a walk-thru of your house and basement and give you a written quote of what you want to do. He will know if its feasible after a walk thru, etc. And then you can see roughly how much it will cost anyways to do.
That's my 2 cents. If you need anything else, please feel free to contact me directly.
Good Luck,

Ken L.

In addition to the other excellent answers, check the zoning. The property may need to be zoned multi-family if you plan to have tenants in the new apartment.

Scott is right. Contact the Building Department in Uxbridge, MA.

Here's the site:
http://www.uxbridge-ma.gov/Pages/Building.php

You'd have to consult your local building department and look at building codes & requirements