Short summary:
Only works on non-med armor (note that non-med armor now gives a LMC-like bonus)
Depending on Refinement type, it changes either max resists or the DCI cap - at the expense of the other.
The types of refinements are specific to the non-med armor types, and drop on various bosses' corpses (champs, boss fights, etc.), in Treasure map chests, and in SOS chests.
They come in various strengths: Defense, Protection, Hardening, Fortification, Invulnerability.
You can get an amalgamator from the Clean up Britannia system, that allows 2 like Defense refinements to be combined into 1 Protection, 3 protection into a Hardening, 4 Hardening in to a Fortification, and 5 fortification into an Invulnerability. For purposes of storage, it's usually a good idea to have a couple of amalgamators handy to combine refinements whenever you get enough to combine. I've had 10 refinements before, for one armor type (1 less than needed for the first 4 categories to combine), that with the addition of 1 Defense refinement, became just 1 Invulnerability through 4 sequential amalgamations, freeing up a lot of lockdowns.
As for their actual use, I've not actually gotten around to doing it yet, but it involves special materials you get from certain NPCs (malleable alloy for metal armor, etc.)