NH has incredibly high property taxes because they have no INCOME tax, so schools, etc are entirely funded by property taxes.
Bear in mind that if you get a job in MA you still pay MA income tax and then NH property tax, so be very careful about doing that.
I would check with a realtor because the taxes do vary from town to town. And, I would caution you that while some towns have very good school systems, in general the NH education is not ranked as highly as some other new England states because they don't have as much $ to spend on the schools. So do your research on school systems as well.
As for how it's paid, you'll likely escrow it quarterly - your mortgage company will figure out how much your yearly tax bill is, you'll pay your mortgage every month and that will include taxes and insurance, and then your mortgage company will pay the tax for you on a quarterly basis.
I agree with another poster - do check with a realtor, because they'll be able to steer you in the right direction. Also, you can sometimes find out what the yearly property tax is when looking at online listings.
Good luck.