A surcingle is essentially a padded, thick belt that fastens around the waist of a pony. It often has multiple attachment points and is an integral part of a pony's harness. This is especially true for cart ponies, where the surcingle often serves as a point of attachment for the cart. A surcingle is designed to take and distribute a reasonably large amount of strain comfortably and safely.
In addition to its use as part of a more complex harness, a surcingle may also be used by itself for ground training (e.g. when longeing a pony). This setup is ideal for short training sessions, when there is insufficient time to fully tack up a pony.
While a saddle girth also attaches around a pony's waist, it is merely designed to prevent a saddle from slipping, and is thus typically thinner and has less padding than a surcingle. It is worth mentioning that the term "girth" is often used to refer to anything attaching around the waist. However, you will usually know from the context whether a saddle girth or surcingle is meant.