Where do you live? - check your country's equestrian federation website for the dressage section. You'll find the dressage rules for your country. Someone may be able to help you with specific federations, I'm in Australia so can only give you the Australian Equestrian Federation website and rules.
As for types of horses, warmbloods (particularly the European breeds) are favourable for those who are experienced riders and striving to be competitive at the higher levels of the sport.
However, for those starting out and in the lower levels, just about any breed will suffice. An uphill, strong conformation and scopey paces will be of great benefit in dressage, making your job and the horses, much easier that if you are starting with a horse that is build downhill and moves flat to the ground (i.e. Your standard QH).
A willing, trainable temperament and eagerness to work is essential, as focus from both horse and rider is needed for long periods of time.
As for what goes into it, dressage is not just 'flat work'. Many eventers/showjumpers etc seem to be under the impression that dressage is just riding around in circles. This is far from the case. We aim to ride and be in control of every step of the horse, have it so tuned into our aids that we merely have to twitch a calf muscle to have the horse respond.
If you're dead serious about becoming a competitive dressage rider and pursuing it for the long term, it requires huge commitment to spend many hours in the saddle, developing the horse's fitness and training. It is an expensive sport to be involved in and far from easy, but well worth it once you 'click'.