Any exercise that involves the use of your stomach muscles and back muscles in a coordinated way counts as a core exercise. For example, using free weights while keeping your core stable trains and strengthens many of your muscles, including your core muscles.
Classic core exercises stabilize and strengthen your core. Classic core exercises include planks, situps and fitness ball exercises.
A bridge is another example of a classic core exercise. To do a bridge, lie on your back with your knees bent. Keep your back neutral, not arched and not pressed into the floor. Avoid tilting your hips. Tighten the muscles in your stomach. Raise your hips off the floor until they line up with your knees and shoulders. Hold for three deep breaths without breaking your form.