There are 29 core muscles that work together to keep the body stable as it generates force, absorbs force, changes direction and moves in multiple different planes of motion. An effective functional exercise program trains your core to do all of this. Rather than training in isolation, we integrate core training into many exercises for other body parts. Ultimately, a core that is stronger while performing movements can improve your overall performance and strength and reduce risk of injury.
If your core training program consists mainly of crunches, leg extensions and back extensions, you're working in only one plane and risk overtraining your rectus abdominus. That's the "6 pack muscle" and yes, you can overtrain it to the extent that it increases the risk of injury - and in a way that will surprise you: A tight rectus abdominus pulls the pectorals forward, tight pectorals pull on the shoulders causing them to rotate in, rotator cuff muscles cannot function effectively and are at risk of injury. Running? By pulling on the chest muscles and moving the shoulders and head forward, an overtrained rectus abdominus can reduce your oxygen intake. Exercises that strengthen the deep inner abdominals, and exercises that incorporate core stabilization with other movements, are an essential component.
Public enemy number 1 is the chair, followed closely by the bench. If you spend a large part of your day sitting, it's a good idea to not to sit or lie down through your entire workout every time you train. Chairs and benches do the work of stabilizing your core as you perform the exercises.