As far as I know, the term fat quarter is only used in the fabric world.
A fat quarter is a size of fabric one can buy.
Most cotton fabric comes around 44" or so wide (it varies).
A fat quarter is the same amount of fabric you would buy in a 1/4 yard, but it is in different shaped rectangle. Rather than being 9"x44", it is half the length and twice the width. Thus, a fat quarter is basically a 1/2 yard cut of fabric sliced down the middle. So, a fat quarter is a piece of fabric measuring 18"x22" (or whatever the length of the particular fabric is). See the diagram below for a visual.
Why would I buy a fat quarter and not a 1/4 yard cut? Good Question!
A fat quarter is often a better shape to work with. A 1/4 yard cut can really limit you. The biggest square you can get out of a 1/4 yard cut is 9". You can get an 18" square out of a fat quarter.
You may have a project you're working on where you don't need a lot of one fabric, but you need more than 9" each way. The benefit (or downfall, depending on your feelings about a fabric stash) of a fat quarter is that you don't have a lot of excess fabric when you end up in this situation.
Note: A fat quarter isn't always necessarily the way to go and sometimes it's not available.
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