Savvy, this is funny because I just posted the training scale on your other thread. Then, I saw this thread, so now I feel silly. But, I gave a link to a great site that explains the training scale concepts.
Rhythm is the base of the scale because it is indicative of relaxation and tension. If the rhythm is bad, common problems are that the walk can be 2 beat (lateral) and the canter can become 4 beats (sometimes evident in upper levers when people try to collect the canter). These things happen because of tension - in his body and/or mind.
In young horses, have you ever ridden one who is new to rider and is getting used to the weight? Sometimes, they rush forward, then go slowly suddenly, then rush forward, etc. Establishing a constant rhythm is really helpful for them. You simply become a metronome for the gaits. The walk is 4 beats, trot 2, canter is a triplet. Transitions from walk-trot-walk and trot-canter-trot are really helpful to help the young horse begin to take weight on his hind quarter and get stronger. Riding 20 meter circles with proper bend are also great ways to help the horse begin to engage the hind quarter.
Mostly rhythm is establishing consistent cadence, either speeding the horse up or slowing him down, and moving forward from your leg. Here is a nice site:The Art of Classical Riding--Dressage Training for Horse and Rider