The process is known as lapping the rings. Lapping being a general term that means to abrade the surface of one or more parts. The purpose is to smooth the surface and/or (in this case for the scope) insure that two mating surfaces (the scope and the rings) are flat and square to one another. Not limited to scope rings, lapping is done for barrel bores, bolt lugs, bolt/action surfaces, etc.
It's pretty rare that the rings for a scope, when mounted, are absolutely concentric and cylindrical. There will be tiny, and in some cases for cheap rings huge, imperfections that cause the rings to be misaligned. The problem with misaligned scope rings is that when they are tightened down, they will put more stress at particular points at the edgges of the rings. A lapped set of rings will have ground these edges down and allow the rings to distribute the stress more evenly around the scope. A good visual example of this is when you see a scope that's been taken off of its rings and you see the ring marks on the scope. The edges of the unlapped rings dig into the scope and wear it down leaving the marks. The stress shouldn't be too bad for a modern moderately priced scope and the main tube body should be able to handle the small stress points of the rings unless you seriously torque down on the rings. It is, though, a big cosmetic issue if you take the scope off and reveal the ring marks. In some rare cases, a seriously misaligned set of rings bolted down tightly will damage the internals of the scope.
Some rings, even the moderate and higher end ones, are pretty tight on the scope out of the box. Not uncommon to have the scope not seat into the lower rings because they are too tight and seem undersized in their diameter. (Have had this happen on a few Leupold rings.) Not only would lapping align the rings in this case but also open them up a little to allow the scope to rest into place easily.
Some folks don't mind the misalignment and feel it's no big deal, the scope can handle it and lapping is not worth the trouble. Personally, I also don't think it is a big deal but I hate to abuse my firearms (and anything connected to them). I hate the ring marks. For just a little hard work you can make a functional setup (unlapped misaligned rings) into a better one. Not satisfied with good enough if it can be made better.
There are kit out there made specifically for lapping rings. Using some sandpaper and a dowel is a bit crude but would probably work. Like sandpaper, there are different grit levels for lapping.
Brownells
Sinclair
