The standing martingale, which I covered in an earlier post about hunter tack, attaches to the bottom of a regular cavesson noseband to keep the horse from raising its head higher than a certain point.
The running martingale is used so much in the jumper ring because it allows the horse much more freedom of movement than does the standing martingale. Because the standing martingale is fixed, there is never any additional give if the horse needs it. Because the running martingale has movement along the reins and is attached to the reins, which also have movement, it is far less restricting. If the horse needs to stretch out over a big jump or requires its head to recover from a stumble, it's easy to give the reins away and allow the horse that freedom.
To use a running martingale, slip the neck strap over your horse's head and attach the martingale to the girth. Unbuckle your reins (at the rider end, not the bit end) and thread one side through one ring, then thread the other side through the other ring before re-buckling the reins.
How tight should the martingale be? That depends, of course, on your horse, but a general rule is to have it long enough that you can pull the straps up to the throatlatch area when the martingale is on.
Make sure that the neck strap is adjusted snugly enough not to hang low and flop around, but not so tightly that the horse will have trouble flexing its neck with it on.
Also be careful not to use a running martingale if you don't have stoppers on your reins. Stoppers prevent the martingale rings from getting caught on the buckles near the bit.
So which martingales are you allowed to use, and when? I'll let the EC rules speak for themselves:
ARTICLE G502 TACK AND EQUIPMENT
1. ... Standing martingales [...] will be permitted in jumper classes where the height of the fences does not exceed 1.15m.
4. In the competition arena:
b) If running martingales are used, they must be unrestricted (For use of
standing martingales, see Article G502.1).