"Independent scholar" is basically a fancy way of saying "not attached to a university or other academic institution." There's no requirement of credentials to call yourself an independent scholar, though there is a professional organization for folks that like to use the label: the National Coalition of Independent Scholars.

An independent scholar is just like a normal scholar, except you have to pay for your own scans at the archives! :) And increasingly academically affiliated people have to do that too, so you know, you're not really that worse off these days... But really, independent scholars turn out good work, they just have a harder go of it, because they don't have access to an academic library and databases (the big one), they don't have a conference/travel budget as part of their compensation, and they don't have dedicated time to work on research and writing. But then again, you don't have to fight a clock on tenure, no one with power over you is telling you that your research topic sucks, and you don't have to deal with the politics.

From the archives/academic library side, working with independent scholars is very common to us, and you are treated no differently than anyone else, your scanning money's as green as anyone else's, your butt sits in the same chairs in the reading room. I can help you find out what academic library resources you can access from your area if you'd like, or this is my general guide to how to library like a pro independent scholar.

"Independent scholar" is the scarlet letter they put on your name badge at the annual conference to let everyone know you don't have a job, and to stay away from you or they might get cooties.

More seriously, as others have said, it indicates someone who does not have formal affiliation. Often this is because they couldn't get a job but are trying to, but sometimes it's because they didn't want a formal academic job and are lucky enough to have the means to pursue their scholarship without institutional support.