With a traditional non-through axle, there's a slot at the bottom of the fork (or the dropouts), the axle is hollow, and there is a skewer through axle. You use the quick-release to loosen the grip around that slot to slide the skewer (vertically) in and out of that slot, while it's still going through the wheel.
With a through axle, there is simply a hole on either side, instead of a slot. You remove the axle entirely out the side of the hub in order to remove the wheel.
Typically, through axles are much thicker axles. The thicker axle is stronger and stiffer. Downhill mountain bikers were breaking axles, and it's a lot harder to break a 15mm through axle than to break a 9mm traditional axle. The stiffness also helps the bike handle better ("more responsive").
From what I've heard from a framebuilder friend, the move to through axles is also because frame manufacturing techniques improved to allow them. With a traditional dropout, if the two sides were a millimeter off, the framebuilder could simply file a millimeter out of the slot on one side to make things work right. Once it's all polished and painted you'd never notice. With a through axle the framebuilder absolutely has to get the alignment just right, because an oval hole just won't work.
The only downside of a through axle is that changing a flat might take an extra second or two.