Transitions within gaits are basically showing the variations of the gait. In trot and canter the variations are collected, working, medium and extended. In walk the variations are collected, medium, free and extended.
First we must understand what these terms mean. When we first start to ride a horse, we use working trot and canter and medium walk. In these gaits the horse is freely moving forward in a good natural balance. The next thing we develop is a slight shortening and lengthening of the stride in trot and canter, the horse is not yet able to perform a collected or medium trot at this point. In order to ride these "forward and back" transitions we try to use mostly seat to stall the horse's movement and mostly leg to drive them forward. If the horse ignores the aid forward, we then do a full transition into the next gait (in the case of the canter, we send them very forward without losing balance on the longside), if the horse ignores the aid back, we do a full transition down. At this point, the only variations in the walk are the medium walk and a free walk on a loose rein, the horse should maintain the same tempo and speed as in the medium walk, but stretch. We also want to encorporate this stretch into the working trot and canter without letting him fall on the forehand.
As the horse develops the ability to lengthen and shorten the stride and frame, we can increase the carrying power of the horse which leads to a collected and medium trot and canter. As we increase the collection (carrying power) the horse is able to perform very collected trot and canter, a collected walk and extended gait in all three movements where the horse extends his limbs as much as possible while staying balanced on the hindlegs.
That is a really brief overview, not sure if it's exactly what you wanted, but there you go! A really good book is "The New Basic Traning of the Young Horse" by Ingrid Klimke, it explains some really good basic exercises for developing the gaits.