A running martingale is similar to a standing martingale or tie-down, except that instead of a single strap to the noseband, it has two, which end in loops through which the reins are threaded (It's possible to jury rig and fit a running martingale as a standing).
The running martingale serves a similar purpose, but is designed not to interfere with the horse throwing his head down and forward and to release if the reins are dropped. Specifically, the running martingale is designed for use on jumping horses.
In traditional fox hunting, most or all horses are put in a running martingale. The neck strap acts as a good "emergency grip" and the rings around the reins serve an important purpose - they prevent the reins from being pulled over the horse's head in a fall. This can end up with the horse putting a hoof through the reins as he tries to get up and can cause a bad wreck. (Which is why hunters in a standing martingale in the show ring is very, very incorrect - but sadly fashionable).
This jumping pony is wearing a running martingale. Note the little leather "tag" next to the pony's cheek - that's a rein stop and is used to prevent the rings from falling forward and getting tangled with the bit or bridle. (Source Malene via Wikimedia Commons).