What Is a Yard of Fabric?
A yard of fabric is cut to be 36" long as it comes off of a bolt (the piece of cardboard that fabric is typically wrapped around). Fabric width has varied over the years, but most of today's quilting cottons measure 42" to 44" across their width (from selvage to selvage). See: Fabric Grain
What Is a Regular Quarter Yard?
A regular quarter yard is 1/4th of a yard, or 9" long and as wide as the fabric -- such as 9" x 44".
Regular quarter-yard cuts can be useful for strip piecing, when long strips of fabric are sewn together and cut into segments.
What Is a Fat Quarter?
A fat quarter is a one-fourth yard cut of fabric that (usually) measures 18" x 22" instead of the typical 9" x 44" quarter-yard cut. The longest side's dimension will vary if your fabric is not 44" wide.
Why Would I Want a Fat Quarter of Fabric?
Look at the illustration and you'll see that a fat quarter gives you the opportunity 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 (less stretchy) 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.
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 the pattern.
A fat eighth is half of a fat quarter and can be cut as shown or in the other direction, parallel to the fat quarter's longest edge. If you're swapping fat eighths, your swap hostess will tell you which measurement to use.