Despite all the talk in magazines, fitness sites, shows like "Biggest Loser" and such about using the core, strengthening the core, core this, core that, I'm convinced that people still don't REALLY know what that means.
Some years ago I remember being told that using your core meant that you suck your belly button in towards your spine. This advice led to countless people sucking in their gut, not activating their back or obliques, and either breathing shallowly or not breathing at all.
All to often I meet someone who wants to "train their core" only to find them doing lots of crunches, leg raises, V-ups and the like. These exercises have their place, but it goes to show that with the ubiquity and the commercialization of the term, "core" has simply been boiled down to mean "six pack muscles".
It might sound silly, but I like to think of the "core" as the "muscle corset". It holds your body in tight, it keeps your spine from bending around, and it helps to transfer pushing power from your legs to your upper body, whether you're holding weights or not.
In this context, think of where a corset holds you. Not just your six pack muscles, but also the obliques, and spinal erectors. They ALL come into play when you need to hold your spine straight, and an over emphasis on the six pack abdominals leaves you in an uneven and unstable position.
Easiest workout to train all the core muscles together: planks. Even if you can do thousands of sit ups, if your other muscles aren't up to snuff, it's going to be tough.
More advanced core workouts: doing exercises with weights overhead. Overhead squats, overhead lunges, or just walking around with weight overhead. I prefer this to doing stuff on bosu balls, because we primarily live on stable ground, so this action is more transferable to real life activities.