Training

I am training a lot. I'm an experienced Tai Chi Chuan (or Taiqiquan, depending on how you prefer to write it) practitioner - I have trained this ancient Chinese martial arts since 1992 and I'm training every day. Tai Chi Chuan is still my "core" style but I have also studied other martial arts styles, in order to become more accomplished - since all styles have their strengths and weaknesses it will never be a good idea to stick to only one style (that will probably prove to be a limitation), by cross-training in multiple styles you will pick up some nice techniques and skills making you a more skilful martial artist. So far I have focused mainly on Mok-Gar Kung Fu and Aikido.
As a part of the martial arts training I'm (like most martial artists) doing a lot of muscular training in order to built and maintain a certain amount of muscular strength, muscle mass and power. My exercising is a combination of body weight training (using no special equipment, only the body weight is being used: push-ups, sit-ups, leg-raises etc.) and weight training. Technique, quickness and flexibility is far more important to a martial artist than muscular strength (and the mental capacities are even more important), that's why I'm not doing bodybuilding since muscles that are too big might be a handicap as a martial artist (they are too heavy and often too slow, often the body will even lack the very important agility as well). But by keeping it at an appropriate level it is serving me well - even though it isn't required (or even an advantage) in Tai Chi Chuan, but under other circumstances it might come in handy. The purpose is to be strong by the pound, not bulky and heavy; in addition to strength and power it is essential to work on achieving stamina, quickness, agility and of course a good technique (as mentioned above) - it is always better to have skills than to rely on brute force.
Of course I'm also doing a lot of stretching - when doing muscular training it is a necessity if you want to keep (or maybe even improve) your agility which is very important being a martial artist. In addition to improved agility stretching has also been documented to increase strength - it is actually possible to built more strength using a combination of muscular training and stretching than by using muscular training alone.
Like most martial artists I'm also running (cardio training). I love the feeling of running (especially when I reach the level of a "runner's high") and it is obviously good for my body and the health in general - and of course it helps staying in a good conditioning. But of course it isn't healthy do overdo it. Running every day (like some people do) will probably mess up the knees later en life - because of the shocks the body (and especially the ankle and knee joints) have to absorb, running is quite tough on the body. The body will need time to recover before being subjected to this kind of "punishment" again - if running is exaggerated you will probably develop a very painful and invalidating knee injury later in life, it could even force you to quit running altogether. Approximately 48 hours of recovery should be sufficient which would imply that you shouldn't run more than 3-4 times a week (or approximately once every second day) - if you require more cardio training than that you should consider doing other kinds of cardio training that doesn't stress the body as much as running does (e.g., swimming or cycling), which will also (as a side effect) help make the training more comprehensive; it is, however, possible to reduce the required recovery time, e.g. by using the ChiRunning method. Of course the frequency isn't the only important factor - you should also look at the training surface since running on a soft and uneven surface (e.g., in a forest or on a beach) isn't as strenuous to the body as running on a hard and even surface (e.g., the pavement in the city), and of course it won't hurt looking at your technique since this will also determine how strenuous the running will be (and you could save your body a lot of unnecessary stress by adjusting your technique if it turned out to be necessary). I used to be a power runner but chose to implement the ChiRunning method since I wanted to be able to run effortless (because of the more relaxed and efficient movements) and less injury prone without having to sacrifice my speed and distance - in fact I'm able to run at least as fast and long as I was before using the power running method, but without the frequent injuries and occasional pain I used to endure. Actually I'm not just running for the cardio training alone, I'm running at a competitive level.

One should always keep in mind that martial arts is meant as a means of self-defense, not a means of aggression. A true martial artist would never train with the sole purpose of being able to hurt or kill other people - I guess no one with a functional brain actually enjoys beating other people up. Of course all martial artists will achieve the skills to be able to do exactly that - but while acquiring these skills the practitioners are also being taught the composure, inner peace, discipline and self-control needed not to hurt anyone unless it can't be avoided, since it is always better to avoid having to fight (if you have to fight you have already lost, as the old saying goes); but if you have to act in self-defense, fighting for your life, you should never hesitate or be nice - in a real fight no rules apply, you just have to get it over with quickly and efficient.
Most martial arts practitioners are not really interested in fighting. They are training to achieve the benefits of the training (e.g., the health benefits, the physical workout and the flexibility), the ability of being able to defend yourself is just an extra bonus. The serious practitioners are also aware of all the aspects of the training (including the philosophy and the mental aspects), not just the fighting techniques.

Tai Chi Chuan (normally translated into "the supreme ultimate fist") is one of the internal styles of the traditional martial arts (TMA). Tai Chi refers to the yin-yang symbol prevalent in Chinese culture and philosophy - more commonly known as the "hard" and "soft" sign, the two opposites coming together in complete harmony. Chuan refers to a boxing method - in this context defined as empty-handed close-quarter combat, rather than a sporting event. This implies that Tai Chi Chuan is a self-defense method based on the Taoist yin-yang principle - it is a system combining the yin and yang approach of serene movements for health and well-being with a strong fighting system for self-defense.
Tai Chi Chuan is often referred to as internal Kung Fu since it is a general rule of Tai Chi that the system emphasizes the development of the internal aspects of the body - breathing, flexibility and the mind - as opposed to external tension or muscular strength.
The principal aim of the training is to teach the practitioners to relax and become fluid in their movements, this will result in smoother actions and quicker response times - the objective is to make the self-defense more of a reflexive action than a repetition of technique.
Chi Kung (or Qi Gong) is a very important part of the Tai Chi Chuan training - and of the other internal styles of TMA. Chi (or qi) is the internal and vital power or energy circulating throughout the meridians - a system very fundamental to the traditional Asian medicine and training systems. Chi kung is developing the inner strength by training and controlling chi.
Other important aspects of Tai Chi Chuan: dim mak (techniques that applies highly damaging strikes against nerve centres and acupuncture points - the use of these "pressure points" is very effective when used by a skilful practitioner) and fa jing (the ability to apply power from within a short distance, which is a prime feature of Tai Chi Chuan). A very important part of the training is toi sau ("pushing hands"), the use of fluid movements consistent with the movements of the opponent, allowing the practitioner to feel and respond to the movements of the opponent - this is a very effective defensive method as well as a means of being able to attack from within a short distance at the same time. The purpose of toi sau is to develop the practitioners awareness, sensitivity, skills and the anticipation of the opponent's movements.
One of the secrets of the world famous Shaolin monks is presumably complete control of their chi - and these monks are probably the world's strongest martial artists, spiritually, mentally and physically (I guess that would imply how powerful chi can be if used well). Tai Chi Chuan and Wushu (translated into "martial art") are styles that are closely related to Shaolin Kung Fu.
Tai Chi Chuan was founded by Chang San Feng, a former student at the Shaolin temple who mastered their system of martial arts. According to legend he travelled to Wutang Mountain (an area populated by Taoist hermits) where he happened to witness a crane and a snake fighting; he was fascinated by the fluidity of their movements and their ability both to evade and attack simultaneously. He adapted his system to incorporate these movements. He was also very skilled at striking the body's acupuncture points, which he also incorporated into his system.
Tai Chi Chuan used as a self-defensive art usually works like this: typically the incoming force will be slightly diverted, the corresponding opening in the opponent's defense will be exploited - the level of response can range from applying locks and holds, to immobilizing an opponent, through to the very effective and painful dim mak techniques (described above). In Tai Chi Chuan the whole body is incorporated into one unit behind an attack. The system includes punches, kicks, locks, open-hand techniques and throws in its repertoire as well as the use of traditional martial arts weapons: sword, broadsword, staff and spear.

I'm also using an mo (can be translated into "press and stroke"), the traditional Chinese Qi Gong massage stimulating chi and often used by Tai Chi practitioners.
Meditation is also an important part of the training.