Yard of Fabric
A yard of fabric is cut 36 inches long as it comes off the bolt. Fabric width varies, but most quilting cottons measure 42 inches by 44 inches. This is called from selvage to selvage.
Regular Quarter Yard
A regular quarter yard is a quarter of a yard, or 9 inches long and as wide as the fabric. Regular quarter yard cuts can be useful for strip piecing. That is when long strips of fabric are sewn together and cut into segments.
A fat quarter is a one fourth yard cut of fabric that usually measures 18 inches by 22 inches. The longest sides dimension will vary if your fabric is not 44 inches wide.
If you look at the above illustration, you will see that a fat quarter gives you the opportunity to cut larger chunks of fabric than what would be possible from a regular quarter yard.
A fat eighth is half of a fat quarter and can be cut as shown or in the other direction.