"So whether you do your first downward dog at 14 or 44, it’s not your history but your presence on your mat that counts."
(K. Pattabhi Jois)
In Vinyasa Yoga you explore the more physical and demanding practice of Yoga. You move with your breath from one pose into the other. Keeping your focus turned inward, you learn to understand your body from the inside out. 
"Developing the mind to become our ally rather than a battle zone is one of the most challenging yet rewarding dimensions of our practice." (Sarah Powers)
Yin Yoga is a practice where you turn inward, stretch your connective tissue while keeping poses usually up to 5 minutes. 
It is a nourishing and healing practice where you learn to delay your reaction to anything that comes up, wether it is in the mind or in the body. You learn to observe what is, without letting yourself get attached. On a physical level there is a lot going on in Yin Yoga. By stretching your connective tissue and/or putting compression on your bones you stimulate the flow of Chi through your body. Chi or Prana is thought to be the vital energy of life.
"When the body relaxes, it literally expands." (Jillian Pransky)
Restorative Yoga is a quieting, supportive and deeply nourishing practice. Poses in Restorative Yoga are mostly passive and done lying on the floor, completely supported by props. It is a practice of learning to let go, letting the body relax thereby reducing stress and tension that's held in the body. Restorative Yoga is an excellent practice to help manage treatment side effects, relieve pain and balance your overall energy. Some examples of conditions that can benefit from a Restorative Yoga practice are; insomnia and sleep issues, post-operative work, diabetic, excessive tightness, digestion problems, back pain and injury, grief, anxiety and emotional trauma, depression, infertility and menopause.