As I was told when I first asked questions on a forum, "the only stupid question is the one you don't ask"
So. Coasters. Well, there are several ways, but the simplest is to buy a craft coaster which consists of two parts, the clear plastic top and an insert. I get mine from Blank Plastic (80p each including postage if you buy ten, less than 70p if you buy twenty) and they are really easy to put together.
OK, you've got your coaster. Now measure the size of the gap that will hold your stitching, and work out how big the design needs to be. So for example if the coaster's recess is 3-1/2" and you work on 14ct fabric, the piece of fabric (once it is cut to size) will be 3-1/2 x 14 = 49 stitches long and wide, and your design should leave a bit of a border, so you could stitch anything up to about 40 stitches square.
When you've stitched your design, iron some Vilene (iron-on interfacing) on the back to stiffen it and keep it from fraying. Turn your stitching upside down and put the coaster's insert on top -- you should be able to see enough through the Vilene to position it the way you want it. Now draw around the insert with a pencil onto the Vilene, cut out the design just inside those pencil lines, pop it into the coaster (make sure it's right side up, because once you've snapped the coaster close it's nearly impossible to open it again without damaging it) and snap the backing into place. Voila, one coaster
The same goes for circular and hexagonal coasters except of course you'll need to make sure the design you stitch is of a shape to fit the coaster's shape.
A completely different way is to hem the fabric you're stitching on (e.g. with four-sided edging) and simply use it as it is -- I have also used PVC glue to make a fabric coaster water-resistant which works reasonably well though they tend to go a bit sticky when they get hot. Ready-made craft coasters are much the easiest option!
Good luck, and do show us when you've had a go!