Dressage is essentially French for training. Every time you ride, you do dressage, at least informally, no matter what saddle you use or discipline you ride.
Formal dressage is more technical. As Spastic Dove said, it's all about you and the horse working in harmony. The horse responds instantly to mere suggestions from the rider, calmly and without resistance (or tack that masks or "fixes" the resistance). For this degree of harmony, horse and rider both need to be in correct form, using themselves efficiently and correctly. In competition, judging consists of several movements individually scored, the higher the level, the more difficult the maneuvers.
Some dressage work would be highly beneficial for you and your horse. It really is the foundation for about everything else. Once you have basic control, you can easily implement basic dressage maneuvers into your flatwork.
A stubborn horse walks behind you, an impatient one in front of you, but a noble companion walks beside you ~ Unknown