Qigong is the art and science of refining and cultivating internal energy.

Ken Cohen, The Way of Qigong

Tai chi is often described as "meditation in motion",
but it might well be called "medication in motion." 

Harvard Women's Health Watch

Qigong and Tai Chi have a long history. They also stem from common origins but have differences in how they’re practiced. The practice of  Qigong stems from the Taoist concept of cultivating qi (chi), primarily for  health practices. Tai Chi also cultivates chi but has more of a  background in the martial arts.

Qigong is a mind and body practice that can improve one’s health and  well-being. Qi refers to the life force or vital energy, which is the  energy that gives power to the mind, body and spirit. Gong refers to  work or gathering of Qi. Qigong practices can include movement, breathing exercises, meditation, sound, self-massage and focused intent. There are likely  thousands of forms of Qigong that have been developed over the years.

Tai chi is an internal Chinese martial art practiced for defense training, health benefits, and meditation. The art is meant to develop and integrate the mind, body and spirit. Developed as a martial art, it is practiced for other reasons such as competitive wrestling in the format of pushing hands (tui shou), demonstration competitions, and greater longevity. As a result, a multitude of traditional and modern training forms exist corresponding to those aims with differing emphasis. 

While both practices entail physical movements, Tai Chi is generally more involved. It comprises a series of many moves, which can take  months to learn and master. There are also shorter forms that can be  learned much quicker. On the other hand, Qigong can be a single movement repeated many  times or a more complex series of movement. Sometimes, there’s no  movement involved at all, but only breathing or meditation.

 The beauty of the qigong approach is that you simply start from where you are.
Tiny steps can still take you where you want to go;
you just need to dedicate some of your time to yourself,
which is a form of self-respect and love

Daju Suzanne Friedman, Heal Yourself with Qigong

Qi Gong

Qigong is China’s ancient system of energy medicine, consisting of exercises and meditations that stimulate the flow of qi, life energy. Designed to restore balance of body, mind, and spirit, it includes exercises for health and vitality as well as “External Qi Healing”, a non-contact energy-field therapy to assess energy, dissolve blockages, and replenish life-force.

Disruptions in the electromagnetic energy of the body occur throughout our lifetime as a result of poor diet, sedentary lifestyle, injuries, surgery, suppression of emotions, and aging. The goal of qigong is to correct these bio-energetic imbalances and blockages. This enables the body to strengthen and regulate the internal organs, the nervous system and the immune system, relieve pain, regulate hormones, and strengthen and release deep-seated emotions and stress.

Medical Qigong is one of the oldest branches of traditional Chinese medicine, predating acupuncture by thousands of years. Medical Qigong includes specific physical movements, breathing techniques, and mental imagery that direct the Qi within the body. Medical qigong therapy consists of treatment by a practitioner to regulate the client's qi. After the treatment the client will be given qigong prescriptions to assist in their ongoing healing. Tailored specifically to the clientmedical qigong exercises use physical movement, breathing methods and mental intention to correct and restore the function in the body.

There is no mystique to Tai Chi Chuan.
What is difficult is the perseverance
It took me ten years to discover my chi,
but thirty years to learn how to use it.
Once you see the benefit, you won't want to stop."

Ma Yueh Liang

Tai Chi

Tai chi is a centuries-old Chinese martial art that descends from qigong. Today, it is practiced as a graceful form of exercise. It involves a series of movements performed in a slow, focused manner and accompanied by deep breathing. Tai chi, also called tai chi chuan, is a noncompetitive, self-paced system of gentle physical exercise and stretching. Each posture flows into the next without pause, ensuring that your body is in constant motion.

Tai chi was originally a martial art, and while nowadays the majority of people practise it strictly for its health benefits, the martial art component is extremely important since it’s an integral part of tai chi. The martial art training consists of several different stages. Let’s look at them and relate them to your tai chi goals. Note that if your goal in doing tai chi is strictly for health, the martial art stages that involve higher risk of injuries are not suitable nor necessary. Tai chi has many different styles. Each style may subtly emphasize various tai chi principles and methods. There are variations within each style. Some styles may focus on health maintenance, while others focus on the martial arts aspect of tai chi.

A growing body of carefully conducted research is building a compelling case for tai chi as an adjunct to standard medical treatment for the prevention and rehabilitation of many conditions commonly associated with age. An adjunct therapy is one that's used together with primary medical treatments, either to address a disease itself or its primary symptoms, or, more generally, to improve a patient's functioning and quality of life.

Links

Qi (Chi) 

Qi Gong & Tai Chi

Mesmerism