A question was raised recently about cultivating “core strength” (perhaps stimulated by wanting to lose those pesky kilos which inadvertently crept on over the holdays?).
I don’t think yogis used the term “core” very much until Pilates came along and made it such a focal point. Then, as there got to be more mention of the bandhas in yoga classes, it was natural to associate them with core strength. We started to “activate mulabandha” and “switch on uddiyanabandha”, even when we weren’t quite sure how to do that.
The muscles that relate to the core are not only in your abdominal area and back, but also in your pelvic floor and hips:  transverse abdominis,  rectus abdominis erector spinae and pelvic floor muscles. Strong core muscles keep your back in good shape, hold your body upright, improve your balance and let you connect with powerful arm and leg movements. A weak core can make you old before your time, as evidenced by the way some elderly people walk.
My favourite text to encourage gradual, safe building of core strength is Asana Pranayama Mudra Bandha, by Swami Satayananda Saraswati.
The section of the book on Pawanmuktasana Series – Digestive/Abdominal Group has exercises that are methodically progressive, challenging in their own way, and lets beginners build foundations for intermediate and advanced poses.

This group of asanas is concerned specifically with strengthening the digestive system. It is excellent for those persons suffering from indigestion, constipation, acidity, excess wind or gas, lack of appetite, diabetes, disorders of the male of female reproductive systems and varicose veins.” – Asana Pranayama Mudra Bandha