“Core stability” is an extremely commonly used term that unfortunately can be overused and misapplied.
What it really means is how to get a strong, steady trunk for our arms and legs to work from. Imagine if you were standing on a broken chair- ie an unstable base. Think how much your body would have to move to adapt to that. Now imagine that unstable chair is your trunk- think about how your body would move to adapt to that “wobbly” base. So the term “core stability” really refers to having a well aligned trunk with good muscle support to hold a good steady posture, allowing the arms and legs to move in a better way.
Lower back pain, sciatica, neck pain, among other things can respond very well to core stability work, once it is done correctly, giving pain relief and also improving the pain in the long term.
Core stability or strength is not only useful in treating pain and stiffness, but in injury prevention, and improving athletic performance. If you think of your body working like a machine- the less energy it uses up by trying to adapt to a “wobbly” base, the more efficient it will move, and the faster and stronger you will be!
See some core stability exercises here: