Whenever stretching comes to mind, most people think deep, long holds to loosen up and relax their muscles but stretching can also involve repetitive movements to help warm the body up and prepare the muscles for exercise. This separates it into 2 different categories - static and dynamic stretching. The simplest way to put it is that static stretching is holding a stretch and dynamic involves movement.
So which should you be doing before or after your workouts? Decades ago, it was habitual to do static stretches before working out but over the years it has been shown to decrease force production, power output, and muscular endurance, which are all pivotal parts of performing well. Imagine going into a deep meditation or sleep and then you suddenly get alarmed. That's essentially what's happening to the muscles when we static stretch prior to a workout. Static stretching, in isolation with longer holds, is best post workout because it helps to cool muscles down, reduces soreness and relax (1).
Dynamic stretching involves movement and is most effective in warm ups to help prepare the body for activity by mimicking the exercises one is about to engage in (1). Doing dynamic stretches prior to a workout helps decrease chances of injury. When we gradually move the body more and more, we build up blood circulation at a pace that doesn't jolt the body and gets it wear it needs to be. Dynamic stretching is also beneficial to increasing range of motion and joint mobility because the lighter versions of normal movements alleviate stiffness so that your muscles can go through their full range of motion (2).
Now that you know the difference, which should you be doing before your workouts? Yes! Dynamic... and after? Boom! Static! Here are a few recommended stretches you can put into practice both pre and post workout.
Dynamic Stretches
Cat/Cow
2. Side to side lunge
3. Standing Twist
4. Arm Circles
5. Pilates Knee Pulls
Static Stretches
Wall stretch - shoulders & back
2. Runner's Lunge - Hips and hip flexors
3. Forward bend - Hamstrings and back
4. Figure 4 - Hips
5. Spinal twist - Back
Sources
Introduction to Dynamic Warm Up. https://www.nsca.com/education/articles/kinetic-select/introduction-to-dynamic-warm-up/
Behm, D. G. (2018). The Science and Physiology of Flexibility and Stretching: Implications and Applications in Sport Performance and Health. United Kingdom: Taylor & Francis.