Most people think of abdominals when they hear the word core. The core consists of muscles of the abdomen, pelvis, low back, and hips.
Core exercises are important in order to maintain balance and stability. Gym equipment is not required to keep your core strong. Any exercise that involves the use of abs and back muscles in a coordinated manner counts as a core exercise.
Example: A simple bridge is a great way of strengthening all of the core muscles. Simply lie on your back with the knees bent, keep your back in neutral spine. Then tighten your abs and squeeze your buttocks as you lift the hips off the floor. Make sure to breathe out when you lift your hips and breathe in when you lower them.