It's a quilters' cut of (usually) cotton fabric like calico or other good quilting fabrics. It's a quarter yard of 44" wide fabric, but instead of being a strip that's 9" x 44" as it would be if you just bought 1/4 yd of fabric off the bolt, a fat quarter is an 18" x 22" rectangle. A piece of fabric 36" (1 yard) x 44" is cut into 4 equal pieces.
Fat eighths are 11" x 18" in case you see those advertised somewhere, too.

