Is there a problem or something wrong with me if I have to strecth and twist my back a lot, because of discomfort or because I just get that feeling?
Back problems are common, and most people don't realize how much they use their back and neck until they become injured. You didn't indicate that you have trouble moving, sitting or standing, and it sounds like you just feel some stiffness, which is really a minor issue most times.
The above poster is very good on medical and science issues, so by all means take her advice and perhaps see a chiropractor or massage therapist. You may also wish to try an over the counter anti-inflammatory drug such as Advil that will often relieve strained muscles and joints.
Also you might want to take it easy for a few days or a week, and don't do any unnecessary lifting or bending. Doctors sometimes advise sleeping on your side (making sure your neck and head are both supported and level). Once you feel better, you can start a regimen of stretching and back exercises to strengthen the muscles there, which will often ward off future stiffness.
It's more than likely that you just have some muscle knots and tightness in the muscles in your back. This can be caused from a variety of things, most notably sitting all day in an unergonomic chair and/or with bad posture. It can also be caused by a poorly supportive mattress that you sleep on. In athletes this can happen from overusage/overstretching of the muscles, or from discontinuing use of muscles once used on a regular basis.
Yoga, a massage therapist, and/or a chiropractor are all things that can help with such muscle knots and tightness. You should try to avoid cracking your back on your own as much as possible because it isn't very good for the muscles in your back. Instead you should do proper stretches, which can help.
Before you endeavour to do any yoga or stretching you should see a massage therapist or a chiropractor, and they can give you some guidance on the stretches etc. that will help.