Due to the complexity of the Tai Chi and Bagua, below is a simplified explanation of their application to the different martial arts and there predominate training methods.
First, the Tai Chi shows the interplay of opposites - mind and body, strong and weak, night and day. The curved line down the middle demonstrates the constant change in the balance of yin and yang at any one time. Together they make up the Tai Chi or the Grand Ultimate. This interplay of opposites is exemplified by the martial art style of Tai Chi Chuan. Tai Chi Chuan took the Yang (ie. physical, strength, aggressiveness, hardness), of external martial arts and added the principles of Yin (ie. internal health, yielding, softness) to create first truly complete or balanced martial art. The essence of Tai Chi is not a style, but a principle that can be applied to any martial art.
The Tri-grams around the Tai Chi are a binary numbering system of the I-Ching. The I-Ching is used to plot and conceptualize the varying amounts of yin and yang at any location in the Tai Chi.

(Ch'ien - Heaven)

Three solid lines plot the point where the Tai Chi is mostly all yang. The sun at high noon is considered at its most yang. Hard physical exercise is also mostly yang. This attribute of being mostly With its attribute of being mostly hard-physical the Tiger~Crane form is represented here.

(Tui - Lake)

The sun is at its most yang state for only a moment then it begins to set. The two dashes represent yin. Thus from a foundation of yang (ie. the heat of high noon), the sun starts to cool (yin). By adding the element of softness to hard-physical the essence of the Northern Fighting Sets is represented.

(Li - Fire)

Midway between noon and midnight the balance of night and day become equal. The heat of the day is the foundation for this point of time with the cooling of night coming after. The martial art of Eagle Claw has a balance of external and internal. From the building of tremendous physical strength internal health is developed.

(Chen - Thunder)

Now, though mostly night, the heat of the day still has some influence. This means despite a movement being 30% hard and 70% soft it is still considered an external movement. The Noble Stances form, with its physically challenging positions (30% hard) is practiced with its focus on breathing and relaxing (70%).

(K'un - Earth)

The opposite of high noon is midnight or the when the day is at its most yin. Where as yang is hard physical exercise that depletes energy - yin is mental internal exercise that replenishes energy. This trigram can be represented by stationary Chi Kung which calms the mind, in turn relaxing the body, and with the right focus turns oxygen into energy to be stored and distributed throughout the body.

(Ken - Mountain)

Midnight, like noon, is but a moment, then the sun begins to rise. Thus from a foundation of yin (the cool of midnight), the sun starts to rise, bringing warmth, yang. By adding the element of physical movement to the internal movement of Chi (energy) one can build health and physical strength without the normal depletion of energy associated with external exercise. Yang Style Tai Chi Chuan and Yi Chuan place mentally moving one's energy before physical movement, therefore placing yin before yang.

(K'an - Water)

Midway between midnight and noon the balance between night and day again becomes 50/50. This time the key point is that the cool of night is the foundation with the heat of day coming after. Chen Style Tai Chi Chuan and the Cheng Style Bagua Zhang have this balance of internal an external. From building and using internal strength external is then developed and exploited.

(Sun - Wood)

Now mostly day, the cool of night has only a little influence. This means that the movement executed using hard-physical was initiated and led by internal Chi. Xing Yi Chuan, Mind Form Boxing, represents this trigram. It is hard and unyielding, and is the most physical of the internal martial arts.

This has been only one method of applying Bagua theory to martial arts training. With further study, more detailed methods of applying the Bagua can be used to define and evolve the use of all parts of the body, fighting applications, and the eight directions of movement.