I don't have one (but would love to have it), but BSR stands for "Bernina Stitch Regulator." It's available on some Berninas and helps keep free motion quilting stitches an equal length. I've definately heard mixed reviews, but if you compare my work on a non-BSR Bernina with my work on a BSR machine, the BSR free-motion quilting stitches are much more even in length, and I can go faster with the stitching looking better.
Hope that helps!
Mattee