AKA a safety collar, limited slip collar, or a greyhound collar.
Martingale collars are a type of dog collar constructed so that the ends of the collar don't meet, but attach to an extra loop of material. This loop sits at the back of the dogs neck, and the leash attaches to it. This allows for a more exact fit, and a more gentle and even pull at the dog's neck. Pulling on the leash will 'pull open' the loop, shortening its diameter, and thus tighten the collar until tension is released.
Martingale collars were designed for sighthounds (i.e. greyhounds and whippets) whose necks are larger than their heads. Because the collars allow for a tighter fit, they keep the smaller-headed dogs from slipping the leash. Martingales are also useful for dogs that like to pull out of their collars.
Martingales are often used as a 'limited choke collar'. Traditional choke collars aren't used much anymore; they're unnecessarily cruel, and the dog can hurt itself trying to get away or pulling at his leash. Instead trainers usually use a pinch collar for a dog that needs more severe reprimands, and a martingale for those that need more than a standard collar, but not as severe as a pinch. The softer the material of the martingale, the more 'mild' the message it sends to the dog.
Although martingales are more gentle than pinch collars, and less likely to choke the dog, they should be fitted by someone who knows what they are doing. If you are using the martingale for training, you should make sure you know how the martingale works, and know about other options (the Gentle Leader or a mild pinch collar, for example) before you start training.