Sides-to-middling a sheet is when the middle of the sheet is worn out through use, you use your seam ripper and rip it straight down the middle (lengthwise), rip or trim off the hems on the sides, flip the two halves, sew the middle together, and re-hem the sides. Then the worn parts of the sheet are on the edges and you have nice strong fabric in the middle. It works best on a full or larger bed, since if it's a single the center seam is kinda irritating.
Since zig-zag sewing machines came in, it's much easier to do! (and at the price for king sheets these days, I'll do it, at least on the most expensive ones. It doesn't show when the bed's made up.)