When referring to the 'core' the reference is not to the '6-pack' of abs that many people are driven to have, but more to the deep set of muscles that lie within the abdominals and the back. These set of muscles attach more specifically to your pelvis and spine and work on movements such as flexion, extension and rotation of your mid-section.
Why are these muscles so important to keep strengthened? Because they stabilize and provide balance to our daily living activities and beyond. From picking up pieces of equipment at work or the gym, core strengthening is essential for prevention of injuries.
There is some great equipment that can be used at home or the gym to work on your balance and strength. By using equipment that provides an unstable surface, your body on that equipment is forced to work against gravity and balance so it won't fall off. Some of my favorites pieces of equipment are (linked descriptions are attached):
Stability Ball http://exercise.about.com/cs/exercisegear/a/exerciseball.htm
BOSU Trainer http://exercise.about.com/cs/exercisegear/a/bosu.htm
Foam Roller http://exercise.about.com/library/blfoamrollerworkout.htm
Wobble Boards http://exercise.about.com/gi/o.htm?
Yoga and Pilates are also a great class to take for core strengthening.
Take the time to incorporate these pieces of equipment into your exercise routine for a great change-up from performing just 'ab' exercises such as crunches. Your benefit will be greater and your body happier!