The best nonprofit membership software depends on what your nonprofit does (a soup kitchen vs an alumni group vs a museum). There are a wide variety of options out there. Some have focused in on the religious space and others on donor management. You probably already know what data you want keep with the profiles. A few years ago as we dug into this question we decided that the most important things to keep track of were donations, events, emails, and then the relationships/contact info.
At the time there weren’t any options out there doing that well, so we built. A lot of options were basically accounting software with some member pieces tacked on. We took the opposite approach and built Fresh Vine to master the day to day life of your nonprofit. There are more nonprofits signing up all the time from all over the place.
In regards to online payments make sure that you ask about rates and fees. Most software services lock you into using their payment tools, or if they support third party options it results in an inferior experience. We did the opposite of that, by partnering with companies that focus on nonprofit payments it lets you shop around for the right combination of rates/fees and services.
It may or may not be a good fit for you. We’d love for you to check it out, and for any feedback people have on what we’ve been up to. Best of luck on your hunt!