Single action: The airbrush allow you to control the pressure you'll have at the "air tip" (in fact most of the single action are kind of on/off you push the trigger and the air flow arrives). This air will pull the paint through a straw which will be lead to a metal or plastic tip (I'll call that a "paint tip") that you can screw up or down to have a less or more paint contributing in a finner or a wider spray (this is a external mixt/bottom feed classic airbrush, there's other but will stick to this one for a start). Most of the time this is good when you want a "same larger" kind of spray. Like a body shell or a big surface to cover... or even little "air line" spray but to change the patern of the spray you'll have to turn you airbrush off and turn (screw or unscrew) the little "paint tip" or you'll have your hand pretty sticky by the end of the process !! OK, you can also bring the airbrush closer to the surface and back and "kinda" control the spray patern but it won't do what a double action will do OK?! The single action is, in my opinion, a charm to clean. They are easy to put together or in pieces because there's less piece in it. The airbrush is quite simple and will let you enjoy this art for a long time because of it's simplicity. Especialy if the "paint tip" is in metal... unlike the Aztek or Testor airbrushes(don't get me wrong here, they are good airbrushes but...). Plastic "paint tip" sometime react with solvent or paint and once the are ruined, you'll need to replace it or you'll have a fuzzy kind of spray and the airbrush will be useless.
Double action: Well... like the name said... it gives you double of the action ! At the trigger, you'll be able to control how much air (the air pressure is often more controlable than the single action) and how much paint you want at the end of the airbrush. Generaly, the Double Action will be internal mixt (it has to be?!) and controled by a slim needle in Stainless Steel and different setting of that needle. When you push the trigger straight down (in a neutral position), very little paint will be push through the "final tip" (or no paint at all depending of the setting) but a lot of air. If you're still push all the way down and pull that same trigger in your way (toward the end of the airbrush), the needle will travel and will let more paint entered the internal mixing chamber resulting in a increase of the spray area because it have the full air flow to work with. So you can play a lot more with your spray, a 50/50 mixt of air/paint, a 70/30 or a 30/70... everything is possible and every setting will give you different result. You're the "spray boss" ! Double action are the way to go when you want to change drasticaly your spray patern while you're spraying. But with all power comes great responsability and the Double action are (compare to a single action external mixt/bottom feed) a pain to clean. Beware of that little needle... it's pointy and have to be... if you give it a tortion or if you bang the tip of that needle, you'll need another one. There's a lot more pieces in a Double thant in a Single action airbrush so be carefull when putting this apart. Double action are mostly in metal and can carry a little more weight than a single action but with all the fun you'll froget this.