Flexibility is defined as the range of motion (ROM) for specific limb/joint. Most of us don't consider flexibility anything special, but, poor flexibility can lead to stiffness, sore joints, poor alignment, and in some cases chronic injuries.
Improving, or maintaining, flexibility is easy. Simple spend a few minutes a couple of times per week moving all your limbs/joints through their full ROM. It is a bit time consuming but you'll feel better afterwards.
Stretching should not cause any pain. Pain is a sign that you are stretching ligaments, tendon, or muscles beyond their ROM. This can lead to injury, scar tissue, and a decrease flexibility.
When stretching you should be focusing on stretching muscles and tendons (the tissue that attach muscle onto bones), not ligaments (tissue that attaches a bone onto another bone). If the stretch doesn't feel good than it probably means you are doing it incorrectly or pushing too far.
When stretching try to hold each stretch for about 20-30 seconds, or longer if you are trying to increase flexibility. Holding it for this long will help stretch the muscles, as they tire a bit the more you hold a stretch.
Be gentle, do not bounce while stretching. There is no benefit to this type of stretching, and it can easily cause injury. The only exception is if your sport requires specific movements which you need to practice. But even then make sure you have warmed up really well before doing bouncing stretches, sometimes called ballistic stretching.
Make sure you are not cold while stretching. Always warm up first, to increase your body temperature, before you stretch. E.g. walk, light jog, etc. Stretching while you're body is cold is simply asking for an injury!
Children, as they are growing, may go through various stages where the lack flexibility. This is normal and occurs because different tissue grows at different rates. Stretching is still good, but always be gentle, do not cause any pain.
Usually girls are more flexible than boys. But there is nothing preventing boys from working on flexibility.
For most people strength building does not reduce flexibility. But, make sure stretching is part of your workout program.
Before explaining stretching techniques it is important to understand that there is an opposing muscle to every contracting muscle. When a muscle is contracting it is referred to as the agonist muscle. The muscle being stretches is called the antagonist muscle. When trying to improve your flexibility you are trying to stretch the antagonist muscle. Remember, it is the muscles we are trying to stretch, not the ligaments or tendons (which will cause injury).
Ballistic stretching is a repeated bouncy stretch. For example, bending over to touch your toes and bouncing just as you reach the maximum stretch. This is the oldest form of stretching. It is effective, but there are warnings that go along with this kind of stretching. You need to be warmed up before doing this, be gentle so you don't cause pain, and if there is pain then you need to take it easy a bit as injury is probable. If your sport involves ballistic movements then you should definitely practice these movements while stretching.
Static stretching involves holding a stretch for at least 5 seconds. The thought is that holding it up to 60 seconds offers the greater amount of improvement in flexibility. Static stretching is just as effective as ballistic stretching and is much safer.
Proprioceptive Neuromuscular Facilitation (PNF) involves alternating stretch then relaxation. The stretching phase will include pushing against a resistance, with the idea of tiring out the antagonist muscle so it will stretch further. For example, lie on your back, have someone lift your leg straight up, have them hold it there while you try to return your leg to it's original position. After about 30 seconds relax, as your partner gently tries to push your leg further up. And repeat. The theory behind this is that the hamstring limits flexibility by resisting the stretch (this is a protective mechanism). But as the hamstring tires it can not resist as much.