You're core consists of the abdominal muscles groups (transverse abdominis, internal obliques, external obliques, rectus abdominis), hip abductors/ adductors, hip flexors, and lumbar spine.
These days I like to preach training for functional strength, and often find that I can't discuss functional strength without discussing core strength and stability. Functional strength can be applied to everyday tasks, and having a strong and stable core will ultimately make these tasks easier. Whenever you are training with any sort of weight, your core must be engaged throughout the session to greatly reduce the risk of injury. It makes sense that training to develop a strong core should be the foundation in any workout regime. Unfortunately, this is a principle that is commonly overlooked.
Here is a quick complex I toss together to really work the core.