A fat quarter of fabric is a one-fourth yard cut that usually measures 18" x 22". To create a fat quarter, cut a half-yard of fabric, 18" along the fabric's lengthwise grain, then cut that piece in half at its midpoint as illustrated.
The longest side's dimension sometimes varies because some quilting fabrics are not 44" wide.
A typical quarter yard is 9" across and as wide as the fabric so a quarter yard of a quilting cotton usually measures 9" x 44". Regular quarter-yard cuts of fabric can be used for any purpose, but are especially helpful for strip piecing, where long strips of fabric are sewn together and then cut into patchwork segments that mimic a portion of a quilt block.
You can use fat quarters for strip piecing by making extra strip sets from shorter lengths of fabric until you've cut the number of segments required for a pattern.
Look at the illustration and you'll see that a fat quarter makes it easier to cut larger chunks of fabric than would be possible from a regular quarter yard, including strips that are twice as long on the fabric's stable lengthwise grain.
A fat quarter offers more versatility, whether it's for patchwork or appliqué. Quilt shops know that fat quarters are popular, and usually offer a wide assortment of them, stacked up and ready to go.
Fat quarters are perfect for sampler quilts, and when you're making a scrap quilt and buying them is one easy way to begin building your fabric stash.
A fat eighth is half of a fat quarter and can be cut as shown in the illustration, or in the opposite direction, parallel to the fat quarter's longest edge. There's no true standard for fat eighths, so ask dimensions before you buy.