This one is the one most people think about when they think of stretching. In this type of stretching you get to a position in which your muscles will lengthen and you hold it for a specific amount of time (15-60s normally). You breathe deeply and slowly allowing your body to relax and lengthen the muscle. Due to the intensity of the stretch, you need to warm up before attempting this type of stretching.
This one is really good at improving flexibility, but as mentioned before it won’t work on real mobility or strength, it focuses only on lengthening the muscle. If solely done, this type of stretch can lead to weak joints.
As the name indicates it involves adding movement to the stretch. So you might move from a low lunge sending your hips back and forth. Allowing the muscle to get to its full range of motion, and then moving back to relax it. You would do this type of movement slowly giving enough time for the body to properly relax and properly lengthen.
This type is the one that most people will do in order to warm up before a work out, because it brings in movement to the joint and muscles. They are great for people that want to gain flexibility but also focus on mobility. The best way to involve your breathing in it is to exhale slowly when you get to the lengthening pose, and inhale when you get back to a more neutral pose.
This type of stretching is a bit controversial. Some people love them; some people hate them. In this type of stretch, you do pulses to access a deeper range of motion. An example would be in butterfly pose (sitting down on the floor, and soles of the feet touching), you would bounce your knees to move them slowly closer to the floor. It is like a dynamic stretch, but there is much more speed to it.
It is controversial because although some people say that by having this small range of motion activated you are preparing your body to go into a deeper stretch. Many agree that during the ballistic stretch you are accessing much more your deeper connective tissues, which isn’t ideal. When stretching you want to focus on lengthening the muscle fibers in the ‘belly’ of the muscle.