For Horses
For Riders

For equestrians, "versammlung" is translated as "collection". Versammlung means meeting. Collect means gather. But both terms have different meanings to the horse person than they do to the general population.  
What is meant by either term is the ability of the horse to increase the flexion of the joints of the hind leg. This increased flexion is like coiling a spring tighter, providing greater power upon its release. The increased flexion causes the hindquarters to lower, putting the horse in a more uphill posture and placing more of the horse’s weight onto the hindquarters. With less weight then on the front legs, the horse because “lighter” in front and more maneuverable.  
This increased maneuverability enables the reining horse to spin and slide.

Western Pleasure horses utilize collection for easily ridden gaits, jog and lope (It’s worth noting that the recent AQHA trend of exaggeratedly low head carriage biomechanically is not compatible with collection).

In dressage, trainers aren’t satisfied with achieving collection, they literally develop it to an art form. Collection is first required at second level, with USDF levels up to 4th and another 4 FEI levels beyond that, each requiring a greater degree of collection.

It is often said that a collected horse steps more deeply under its body, however that isn’t actually true. The difference isn’t so much in where the feet land as where they take off. Let’s look again at this series of photos from Takt, specifically at the right (closer) hind leg:
If this horse was collected, it would have picked up the foot before frame 6. In a collected horse, the pushing phase is much less, and the carrying phase is more pronounced. The strides shorten because the horse is picking his feet up more quickly, the feet have more “air time” and the tempo is said to become more “cadenced” or defined.  

In frame 1, the foot is about to land.
In frame 2, it is weight bearing in front of the vertical.
In frame 3, it is vertical. This is the point at which it bears the greatest amount of weight.
In frames 4, 5, and 6, the foot is grounded and behind vertical. From this position it pushes the horse forward.  
The lightening of the forehand unburdens the shoulders of the Saddleseat/Park horse for high-stepping, expressive movement.
Collection empowers the jumper with adjustability of stride and quick turns.
Once the horse has developed the strength for collection, it can be introduced to the curb bit, whether the curb is eventually used by itself as in western disciplines or as a part of a full (aka, double) bridle as in English disciplines.

It isn’t until a horse reaches this stage of his development that he has strength enough to respond to the curb bit correctly, for the action of the curb to go through his whole body and increase the flexion of his hind legs, as is its purpose.  If the horse can't do everything you require of him in balanced collection in the snaffle, he is not strong enough to carry the curb.