A fat quarter is a ¼ yard or in this country ¼ metre. It should measure just about 20 inches along the selvedge and about 22 inches on the crosswise grain. Imagine a square cut horizontally then vertically to make four quarters. A thin quarter will measure 25cm by the width of the fabric.
A quarter metre, cut from the bolt is a quarter metre and should measure 25cm x width of fabric.
A fat quarter is a half metre cut from the bolt and then cut lengthways down the middle and so should measure 50cm x width of fabric. If the fabric is 112cm/44" that will be 20" x 22".
Imperial FQs, as used in America, are half yards cut from the bolt and cut down the middle, the resulting size typically being 18" x 22". In UK everything (except milk and beer!) has to be sold by metric measurements. Whilst it's not illegal to sell 18" of fabric it should aslo be marked as 45cm. Strictly speaking if anyone in UK is selling Imerial FQs they should be very clearly labelled as such and the metric measurements included, but that isn't always the case. If a pack of FQs seems cheap check the sizes!
I think that the reason why quilters, or rather patchworkers, use inches is because it's so big in USA and most of the books etc are American and instructions are in inches. Also the quarter inch seam allowance is generally easier to work with than 6mm or 6.5mm.