The brain (cerebrum) is divided into two halfs, which are connected by a bridge (pons) of nerves. This connection is larger and denser in females which means that women and girls (usually) have naturally more balanced brain hemispheres.
Because the strand of nerves crosses over at the level of the neck, the left brain (male) governs the right side of the body; and the right brain (female) rules the left side of the body.
The left brain is dominated by "male" (yang) qualities, such as analytical and logical thinking, categorisation, aggression (or power/drive) and egoic and seperatist thinking. The right brain is responsible for "female" (yin) thinking patterns, such as creativity, integrative and associate thinking, meditative and mystic states, the resolution of bounderies and a perception of oneness.
We meed both brain hemispheres in order to be able to function as humans in a three dimensional world. In patriarchal societes the left brain and masculine qualities are (over)emphasized. You can see what happens if the left brain hemisphere stops functioning completely in the excellent book "My Stroke of Insight" by neuroscientist Dr. Jill Bolte Taylor.
In yoga, male energy (the sun), is referred to as Shiva, and is the form (matter) of the universe. Male energy expresses itself through the right side of the body. Female energy (the moon), is referred to as Shakti, and is the force of the universe. The female energy is the domain of the left side of the body.
It can be very interesting to think about our organs and their position in our body. The heart, for example, is on our left side, and as such has feminine (yin) energy. The liver is on our right side and has masculine (yang) energy.
As one person we need to be in constant balance to be able to achieve the optimal state; mental, physical and emotional. Both sides, male and female have to attain a harmonious whole.