This is definitely NOT a stupid question! I think it's really important for riders to ask questions like this to understand the WHY and HOW of everything they do with horses. Asking lots of questions helps to make good horsemen and women.
A running martingale is a bit like a training yoke. It often has a strap that goes around the horse's neck, in front of his withers. Then it has a strap that runs through a hole at the bottom of the neck strap just above his chest at the base of the neck. One end of this strap is a loop that the girth is threaded through. The other end is split into a "y" and has one ring at either end. The reins run from the bit through the rings to your hands. (You can also get a running martingale attachment that clips on to a hunter-style breastplate.)
The purpose of this piece of tack is designed to give the rider more leverage to control a horse who gets his head up in the air and his nose on a horizontal plane. It gives the rider more control when needed, but is inactive when not needed. For example, if you are riding a young horse--he gets frightened and raises his head in the air. If he's wearing a running martingale, it will activate once his head gets above a certain point, giving you the ability to get him back under control quickly. And it's different from draw reins in that it doesn't PULL your horse's head down, but it does correct the horse.
You will often see a running martingale in jumpers or on eventers. A standing martingale--one that has a fixed strap between the noseband and the girth--is not allowed in jumper competition. I'm not sure about the eventing rules, but my guess is that only a running martingale would be allowed in the eventing or show jumping phases. A standing is too restrictive for what is being asked of the horses in those disciplines. The running gives the horse more freedom of his head and neck for the big efforts that are required. Running martingales are also legal for warming up at a dressage competition--whereas drawreins are not. It will have to be removed before going into the show ring, but it does provide a handy training tool if needed.
A couple of safety tips: 1. If you use a running martingale, make sure your reins have "stops" on them. These are adjusted a few inches from the buckle where the reins attach to your bit. These stops ensure the rings do not get caught on your rein buckles, which can have dangerous consequences. 2. In the same vein, if you are using a full-cheek bit, make sure you have keepers on the bit. There is a danger of the martingale rings getting caught on the full cheek, again with disasterous consequences--one of which could be a horse flipping over on you. A safer bet is not using a full cheek at all in conjunction with a running martingale--keepers or not.
Hope this helps. Let me know if you have any other questions about this or other items!