Most people think of the core as a nice six-pack, or strong, toned abs, but the truth is that the abdominal muscles are only a small part of the "core muscles". The major muscles of the core is actually the entire torso area of the body -- this includes the abdomen, mid and lower back, and peripherally, includes the hips, shoulders and neck.
The strength, speed and ease of your day-to-day bodily movements, or any other physical activities, especially in sports, is vastly determined by the strength of your core. To build a strong core you need to exercise a variety of muscles from your hips to your shoulders.
I'm going to quickly show you here, one of numerous ways to challenge your core muscles (with and without the stability ball) -- including progression from beginner, intermediate, to advance. Try it!