The term "core" refers roughly to the muscles that connect your ribcage to your pelvis. There is no single, universally accepted definition, but the core always includes the surface and deep abdominals and lower back muscles, and often includes the lats and serratus, and sometimes the glutes and hip flexors.
Why train your core? Fundamentally, the core transfers energy from the lower body to the upper body or vice versa. Imagine for a moment that you had no core muscles. What would happen when you turned your hips, say to drive a punch? Well, the hips would turn, your spine would twist, and your shoulders would stay right where they were. There would be two problems: first, your punch would have very little power. Second, your spine would be in big trouble - spines are meant to twist, but only a little, and not very vigorously. So bad punch and bad back.
Now nobody actually has no core muscles - when somebody says they have "no abs" they mean they're too fat to see their abs, not that the muscles are actually missing. But many people have weak core muscles, and they end up with sloppy technique. You can see their hips move, but the ribcage isn't totally in sync. Basically, they're losing energy in their hand techniques because some of the hip drive isn't transferred properly. Again, bad for the punch, and bad for the back. Ideally you want your hips and ribcage to move together as if there were iron bars attaching them. Want a good visual of this? Watch Enter The Dragon and watch the scenes where Bruce Lee is fighting a bunch of no-name bad guys. Watch him punch them - every time his hips twitch his whole upper body moves perfectly together. There's a reason the guy was such a fanatic about ab training, and you can see the value of having super strong abs in his movement.
So how to we train the core? Let's think about what the core does. Basically, the core flexes the trunk (brings your chest forward), extends the trunk (brings your chest backwards, like when you arch your back), twists the trunk both ways (like when you throw a cross with either hand), and bends the trunk side to side (I'm sure there's a fancy word for this but I don't know it). Or, you could say it does all those things for your hips, if your upper body is stationary - so if you hang from a chinup bar or something, the core moves the hips in all those ways. Now here's a mindbender - since the core flexes the trunk, it also works to prevent extension. In other words, if some force is pushing your upper body backwards, the core is what allows you to maintain your alignment. The same is true for rotation. Imagine throwing a medicine ball with one hand. the reaction force of that medicine ball pushing into your shoulder tries to twist your trunk away from the throw. Your core prevents that from happening.
Most people train the core with lots of crunches and leg raises. That's trunk flexion - crunches flex the trunk, during leg raises the weight of your legs pulls your trunk into extension (your back wants to arch) and your core has to prevent the extension (preventing extension is roughly the same muscles as flexion). That's not a terrible start, but we can do better.
Here are the bullet points:
- Pick exercises where your core is working to keep your posture neutral, not ones where you are moving the spine. Planks are better than crunches. Throwing a medicine ball is better than that exercise where you lie on your back with your arms out and legs sticking straight up and rotate your legs from side to side. There are two reasons for this: First, the core muscles work harder when resisting movement than when they initiate movement (sounds weird, but it's true). Second, rotating or flexing the spine a lot is asking for trouble - as in back injury trouble.
- Work exercises where you resist extension (planks), resist rotation (one armed planks, birddogs, one arm medicine ball throws), resist yaw (side to side movement - like side planks), and resist flexion (deadlifts, swings). The exercises can absolutely be ones you already to for other reasons. Swings aren't "core" exercises, but they work the core. That's fine. Just make sure to add core dedicated exercises if you aren't incorporating a challenge in your other exercises.
- Progress to movements where you are resisting that motion in unstable ways. Planks are great. Try doing them on an underinflated exercise ball (Swiss ball) instead. Don't settle in - jiggle around the ball so your core is challenged by a rapidly varying force. Try carrying around a slosh pipe (a 8-10 foot pipe, pvc or something, half filled with water). You'll get a better contraction and you'll prepare your core to stabilize you in a dynamic environment - like a sparring session.
- Don't settle for doing looong sessions of moderately challenging exercises. There's no reason to hold a front plank for five minutes. Once you can hold it for one or two minutes progress to something harder - ab wheel rollouts, swiss ball rollouts, superman planks, something hard. Treat your core strength like your strength anywhere else in your body.
- Don't think that because the core is so important that you should make core-specific training the bulk of your workout. Working only the core is as silly as working only the legs in isolation. You can't actually punch or kick anybody with your core!
- Don't exhaust your core early in your workout! The last thing you want to do is really fry your core muscles early in a routine, then throw a bunch of punches or kicks later. You need your core for all that other stuff! Do your core last or close to last in your workout.