Martial means "military" and art is a means of expressing one's inner self. Martial arts is for many people a way of expressing themselves through the disciplined movements they learn from their style or styles. True martial arts is not about dominating an opponent, but rather mastering oneself.
There is a big difference between training and learning (or instructing). When we are learning a skill, we are teaching our bodies to do as our mind commands. When we are training, we are conditioning our bodies and minds to improve on a learned skill. You cannot train on a skill which you have not first learned. We learn from many different sources throughout our lives. Teachers, books, friends, videos, events which we witness or hear about, all these and many more teach us valuable lessons about every facet of our lives.
In the martial arts there is a saying that goes "We forge our bodies in the fire of our will." We learn skills which teach us to unite the mind and body, thus allowing us to progress much further than would be possible without this unification. Correct repetition of a movement will eventually bring about a higher degree of skill in that movement. It is generally a good idea to have end in mind when you start out. For example if I am to learn to do a side kick I want to have the ultimate side kick in mind when I start. I must work diligently to improve my side kick until it matches my mind's idea of the ultimate side kick. Along the way I will likely discover the need to have greater flexibility in order to achieve my goal. I may also decide that it is necessary to strike something in order to bring the level of power up to where it needs to be. There may be many other things which I will discover on my way to the ultimate side kick. In order to achieve success I must start out with the end in mind and then use my will power to train my body to do what I want it to. Self-discipline is probably the most important thing we can gain by training in the martial arts. The hardest things to do in life are the things, which we must make ourselves do, even when we don't really want to. Self-discipline could be described as using the mind's will to overcome the body's weaknesses.