I think there is alot of confusion about what the "core" is also,when and how to train this area. Here is my take on the topic.

What is the "Core"?

The "Core" is the Lumbo-Pelvic-Hip Complex--it is made up of approx. 29 muscles that include: deep abdominals, obliques, deep spinal musculature, hip flexor group and the glutes.

--A misconception: Most people think of the core as only the superficial ab muscles..... the one's you can see in the mirror.

The core is where the human body's center of gravity is located and where all movement begins. Proper development of the core musculature will result in more efficient movement patterns and optimal performance. Efficient movement equals less chance of injuries!

When should I train the Core?

It is important to remember that the core musculature is the start of all movement and therefore makes sense that it is developed to its optimal level and should be performed at the beginning of your workout.

--If your core is weak and/or untrained it leads to inefficient movements that could lead to injuries or patterns of injuries.

How should I train the Core?

Core training provides intrinsic stability(think of an internal weight belt) to the Lumbo-Pelvic-Hip-Complex, which allows for optimum efficiency of the rest of the body.

-Core training should emphasize the entire muscle contraction spectrum:

1.) Stabilization--Isometric Contraction(maintain a certain position for a period of time)
Examples:

Planks
SB(stability ball) Push-up Bridge/Walkouts
SB Hip Bridge
SB Cobra

2.) Force Production--Concentric Contraction(shortening the muscle against resistance)
Flex/Ext/Lateral movements
Examples:

SB Curl-ups
SB Back Ext
Sit-ups

3.) Force Reduction--Eccentric Contraction(elongating the muscle against resistance)
Examples:

Roll-outs
Reverse Sit-ups
SB Jackknife
SB Rev. Curl-ups

4.) Rotational Force--Rotational/Diagonal Patterns
Examples:

MB(medicine ball) Twists
MB Diagonal Chop/Lift
SB Skiers
SB Arm through's

5.) Neuromuscular Stabilization--Balance
Neuromuscular Stabilization enables the body's neuromuscular system to synergistically produce force, reduce force and dynamically stabilize the entire kinetic chain(body) in all 3 planes of motion. Neuromuscular efficiency enables a person to maintain balance during functional movement patterns whether on the field of play or in day to day activities. In addition Neuro. Stab. increases your coordination and decreases the chance of injury. Balance is a component of all movement whether strength, speed, skill or flexibility dominates the movement in question.
Examples:

1 leg squat, touch & reach
1 leg press, toss & catch
Dynamic single squats(3 planes)