I made these in university when I had a bunch of dud CDs/DVDs, a lack of coasters and some free time (ahh life of a student). Easy to make, and endless variations possible.
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Step 1: Supplies
CDs or DVDs that you can sacrifice
micaceous iron oxide (dark gray acrylic colour, with some sparklies) - you can use any sort of acrylic colour really but this one struck my fancy
Paper Mod Podge
(optional) lacquer or acrylic sealing spray
Step 2: Sanding
Sand off the top shiny layer so the acrylic can adhere to the surface better. You can do it lightly so the base still shows a complete shiny reflective surface, or take the layer off completely to show the acrylic through the bottom (I did the latter but left some on at the edges). You can also etch the surface with a knife (carefully) to create patterns.
Wash off the residue with water - be careful as some of the reflective layer could be in tiny splinters instead of dust.
Step 3: Acrylic, Decoration & Sealer
Spread the micaceous iron oxide acrylic over the top surface of the disc in a thin layer. This stuff is thick enough that you can either do it smooth or texturize it. If desired, place paper graphics or other decoration on the acrylic before it dries (it acts as an adhesive as well). Best to use something flat as you'll be balancing a cup on top.
Let dry completely (the thinner the layer, the shorter the dry time).
Once dry, coat with a thin layer of Mod Podge to seal - it goes on white and dries clear. This is supposed to be non-tacky, but I find that it is just a tiny bit sticky still - at least it will follow your cup around! You could use a clear acrylic spray instead - I tried varnish before but it has a yellow tinge.
Step 4: Alternative: Lacquer or Acrylic Spray
Another way you can do this is to use some lacquer or acrylic spray in the colour of your choice - use this outside or with good ventilation, and on top of newspaper/garbage bags. Apply a thin coat (according to spray can instructions) and let dry completely - it will adhere to the shiny layer and dry smooth, but where it is on the plastic itself it will dry in a cracked pattern (kind of like those nail polishes). If you just use one coat, some of the original colour of the CD/DVD will bleed through in the exposed plastic part.
Try not to overspray, or some of it will leak underneath the disc (as pictured). >_>