I'll define here the foundation of a yoga pose as any part of the body that is in contact with the floor and that is supporting some portion of the body's center of gravity. In general this means that any part of the foundation that presses down into the floor because it is supporting the weight of the body.
Part of staying balanced can include choosing where to position your center relative to your foundation, i.e. which part of the foundation are you trying to keep your center over, and also being able to sense where you center of gravity is relative to your foundation.
This means not only sensing when your center is over your target area but also sensing when your center is moving away from the target area.
Sensitivity without the ability to effectively respond is an exercise in frustration (unless you like frustration) and so another part of balance is being able to control your body in such a way that you can position your center and move it as required.
One idea that may be helpful in this regard is to think of each part of the body, the forearms, the upper arms, the head, the ribcage, the pelvis, the thighs, the calves and feet, as each having their own center of gravity.
The position of the body's overall center of gravity depends on the position of all of these elements relative to each other.
As an example, standing upright with arms by your sides places your center of gravity somewhere on the centre line of your body, over a point between your two feet.
Standing upright and pushing your hips left and leaning your torso right then your center of gravity will remain stationary relative to your feet. Instead your pelvis will have moved relative to your center so that your center is now to the right side of your pelvis. (See pictures above).
Bending forwards at the hips while upright, your hips will move back and your torso forwards. If you lift your heels (while staying balanced) then your center will be at the point of your body directly over your foundation.
In this position if you reach start with yours arms back and then reach your arms forwards, your body will move back relative to your feet. Reach your arms back and then your body will move forwards. What this means is that as your arms move, you shift your bodies center relative to your pelvis.
In this case your center remains fixed relative to your foundation but your body moves relative to it.
Moving into an arm balancing yoga pose like Crow Pose (bakasana) you start with your center of gravity over your feet.
For more on Crow Pose you can read:
Since your hands are in front of your feet (with your knees against your arms) to balance on the hands in this pose you have to shift your center forwards, so that it is over your hands. So you have to move your body forwards.
This same is true in most arm balances if you start with your weight on your feet (or foot) behind your hands.
For more on Balancing in Yoga Arm Balances read:
To balance in these yoga balance poses you have to get your weight, your center of gravity forwards over your hands.
In a pose like half moon pose it can be easy to shift your weight towards your hand so that your center is actually over a point between your hand and foot. How do you then lift the hand? By shifting your body so that your center of gravity is over your foot.
For more on balancing in Half Moon Yoga Pose:
To balance in side plank pose, you have to have your center of gravity somewhere over the line that connects your supporting hand and foot. To make it easier to keep your center of gravity over this Foundation Line you can make your foundation wider by placing one foot in front of the other. To balance with only one foot on the floor make sure that your bottom foot is stable.
One of the easier ways to stay balanced in this pose is to move the hips. If you feel your weight shifting forwards then move your hips backwards. If you feel your weight shifting backwards then move your hips forwards to compensate.
And for either of these corrections stop when you feel your center of gravity is over the center line otherwise you'll just hip sway between the two nearly unbalanced positions.
For more on learning to balance in Side Plank Yoga Pose:
Balancing In Side Plank.
For some fun (and strengthening) Side Plank options:
In Headstand yoga pose you can keep your center over the line midway between elbows and crown.
For a step-by-step guide read
How to Do A Headstand.
If you are trying to lift into headstand with legs straight then it helps to understand that with your feet an inch off of the floor you have to move your hips back to counter balance the weight of the legs.
Moving the legs to horizontal the hips have to move back even more.
From here approaching the legs vertical position the hips can move forwards so that you keep your center of gravity over your foundation.
For more on lifting into headstand with knees straight read
To avoid excess stress on your neck move your hips forwards (away from your head) to balance the weight of the legs. As your legs approach vertical you can move your hips back, towards your head, to keep your center of gravity over a point between shoulders and elbows.
An alternative is to prebrace the neck prior to lifting the feet by pressing the back of the head into the floor.
This then "widens" or broadens your foundation from front to back since the tension in the back of the neck forces the head to act as part of the foundation. You can then lift your legs without having to move your hips so far forwards.
For balancing in handstands the intent can be to keep your center over your hands. When jumping up you'll probably have to angle your arms so that your shoulders move forwards. This then gets your center over your hands.
As you bring the legs up past the horizontal then you can bring your shoulders back over your hands so that your arms are vertical.
In both headstand and handstand to reduce the need for your center of gravity to go forwards, bend the knees or move the legs out to the sides as you lift them
Or scissor the legs, bringing one leg behind the body to help balance the weight of the leg that is in front of the body.
If you are doing a handstand against a wall then you can use a similiar understanding to get your feet off of the wall. Either pull your pelvis forwards first to get your center over your hands or scissor one leg forwards to get your weight over your hands. If you pelvis is forwards, then as you bring your feet off of the wall move your pelvis backwards to maintain balance.
If scissoring the legs, then after removing your other leg from the wall bring both legs together to keep your center of gravity over your hands.
I've included a slightly more detailed discussion of balance, (most notably using floor contact to feel where your center of gravity is with respect to your foundation) in the ebook Balance Basics.
I've included more detail on stability and balance in improving balance.