Side stretches stretch the side of the body. Depending on how you do them, they can also strengthen the side of your body.
Side stretches can work on the hip and all sections of the spine.
When side stretching the spine it seems (or may seem) pretty obvious that after stretching one side you stretch the other. You can stretch the sides of the lumbar spine, thoracic spine (the part that the ribs attach to, and thus the ribcage) and the cervical spine (that connects the head to the ribcage) individually or together.
If you consider the side of the hip to only be the outer part, then for balance, you also need to stretch the inside of the hip.
A simple standing side stretch is Standing Side Bend.
The basic approach to this pose is to stand with feet hip width to shoulder width apart with feet parallel and knees straight. Push your hips to the right and bend your lumbar and thoracic spine to the left.
To make it easier to push your hips to the right use your left foot to push the floor to the left.
So that it is easier to breathe smoothly I'd suggest holding the basic shape of the pose and then slowly push your hips further to the right and then relax. As you push your hips to the right inhale, and as you relax exhale.
Then once you are comfortable with this action focus on reaching your ribs away from your pelvis.
So first push your hips to the right, then reach your ribs upwards and to the left, away from your pelvis. For an additional stretch for the shoulders and arms, reach the arms upwards and to the left, past the ears. As you reach your ribs away form your pelvis, you can also reach your arms further away from your ribcage. Do all of these actions sequentially while inhaling (hips, ribs, arms) then relax them while exhaling.
Another standing side stretch that can be even more effective for both stretching the side of the waist and strengthening it is Triangle Yoga Pose.
The variation that I use is slightly different than the way a lot of yoga teachers traditionally teach it. Rather than resting the bottom arm on the leg or a yoga block or on the floor, I have my students lift the bottom arm.
First, to get into the pose step the feet about a leg's length apart. Turn the left foot inwards slightly and turn the right foot out 90 degrees. You'll be bending to the right.
To bend to the right drop the right hip so that the pelvis tips (or tilts) to the right. Then rest your right hand on your shin. Position your ribcage over your right leg.
Gradually bend your right elbow so that your ribcage sinks down, towards your thigh. Keep your ribcage over your leg. From there gradually relax your right hand and then when it is fully relaxed, lift it off of your shin. This can be an inhale. Then put the hand back down, use your arm to support your ribcage and exhale.
Repeat a few times so that you are comfortable with the hand-lifted position, then reach the right hand to the left after lifting it. Inhale and reach ribs and arm to the right, then exhale and relax both actions gradually. You can let your ribcage sink down. Repeat a few times, then add the top arm, so that both arms reach to the right.
When you are finished try to smoothly and slowly stand with your arms continuing to reach past your ears. Reach upwards at the top then repeat on the other side.
Another way to stretch the side of the waist is while doing a standing forward bend.
If you aren't comfortable or flexible enough for standing forward bend you could leave this pose out or you could rest your hands on the edge of a table, chair or pair of yoga blocks.
Bend forwards and place place both hands on the floor.
To bend the waist to the right:
You may find it helpful to reach your left hand further to the left, and perhaps backwards towards the heel or past the heel of your right foot.
A simple way to stretch the side of the body is while sitting with the legs crossed. You could also kneel.
Bend forwards, reach your hands forwards then bend your ribcage to the right, reaching your arms to the right also, to stretch the left side of your waist. You can deepen the side stretch by bending further to the right and/or by reaching the arms away from the ribcage, particularly the left arm (if bending to the right.)
Use your hands (against the floor) to help pull your ribcage to the right.
If you are comfortable in double pigeon pose, you can also bend forwards and side bend towards either the top-leg side or the bottom-leg side with the legs in double pigeon.
Another way to side bend in pigeon pose is to have have one leg forwards and the other leg back in pigeon pose glute stretch with the front leg hip on the floor and the front shin parallel to the front of your mat. From here bend towards the front-foot side, trying to reach your nose, face or same-side shoulder towards the front foot.
Other leg positions you can use for this seated-forward-bending-side-stretch include the easy bharadvajasana foot position.
Another side stretching option that you can use with the easy bharadvajasana foot position is to sit upright and bend towards the kneeling leg side. This can be used as a preparation for reclining hero pose or triang muka ecka pada paschimottanasana, a pose found in the ashtanga yoga system. It can also be a way to stretch the front of the ankle and even as a quad stretch.
Some of my favorite side stretch variations are shown below.
The one I currently use quite a lot is the seated wide leg side bend. Variations of this include bending the other knee so that the shin is folded to the outside of the thigh (hero position) or to the inside of the thigh.
For all of these side stretches you can bend towards the straight leg and let your ribcage sink downwards.
Support yourself with your bottom arm and gradually bend the elbow to let your ribcage sink down. To add weight as in triangle pose, slowly lift the arm, reach the ribs away from the pelvis, then gradually relax. Initially put the arm down when relaxing, then try to keep the bottom arm lifted for both the inhale and the exhale.
From there, you can reach the top arm over the head. This can add some weight to the stretch.
After some practice you may be able to grab the foot with the top arm over the head. For this position you can press the leg down and then pull up with the arm (while continuing to press the leg down) each inhale. Then relax both actions while exhaling.
Half hero side bend is similiar to the previous side stretch except that the knee of the leg you are bending away from is bent with the shin folded to the outside of the thigh.
This foot position makes for an easier side stretch, particularly if you are interested in grabbing the foot. It is easier because the foot position makes it easy for the bent knee hip to lift.
If you do grab the foot, one option is to start sinking the bent knee hip so that you increase the side stretch at the waist.
Another option (you can do both) is to pull upwards with the arm while resting downwards with the straight leg.
Here the idea is not only to create tension (which gives sensation and proprioception) in the leg and arm, but also in the upper side of the torso. You may find that you can deepen the side bend while maintaining the tension in your arm, leg and side of the torso.
In the janu sirsasana side bend the shin of the bent knee leg is folded to the inside of the thigh. This foot position can actually make it more difficult for the bent knee hip to lift making this is a more intense side stretch.
Note, if you sit up slowly, and in control from all of these side bends, and without using your arms to help, then you also exercise the muscles that you have just been stretching.
As a yoga teacher, I'm constantly exploring new exercises, new ways of doing yoga poses.
There is no single "right way" of doing a yoga pose. Instead, there are options. And the better you are at "feeling" your body, the better you can get at choosing the right option for your body as it is now.
For any technique, the point of practice is to learn feel it and to control it, so that it can be used without thinking about how to use it.
And that is more or less the approach taken in all of my ebooks and videos. They help you to feel your body and control it so that you can work towards using it effectively in anything that you do.