We just finished using a Blue Bamboo unit, FYI. We paid for the unit itself and then had monthly fees for the transactions. This was about 5 years ago before square readers were on the market and perfected. We were taking payments by phone on occasion (manually punching in the numbers on the unit which looks like a calculator) but also "live" payments from fairs and events we attended where we physically swiped the card.
There's a monthly fee from a credit card processor (such as AuthorizeNet but there are others) and then there's usually another institution that gets the money and puts it in your bank account (Chase Paymentech is one of many). You can tie these into your regular bank account but be sure you investigate all the fees. There's a monthly fee, first of all, but then there is a per-transaction fee. If they try to take money out but there isn't enough, of course there's a fee, but if they try to put money in and there's any kind of a glitch, they take a fee for that too!
You'd have to investigate whether the square reader goes on any phone or if you need a specific phone issued to the non-profit. Start with your own bank where your group has the account, and ask about everything!
You could also look into Pay Pal - but sometimes people get confused and think they need to have a Pay Pal account themselves. They don't. Pay Pal can act just as a credit card processor. We got rid of it because too many customers got confused by it and we weren't big enough (like, say, Amazon!) to be able to handle those who gave up out of confusion.
A lot of it has to do with how well your website is designed and how clear things are. If you don't have a good website, you'll need to look into design and hosting fees for that as well, unless it's already done.