When it comes to strengthning the "core" your speaking of working parts of your body that keep your spine stabilized.
I'm not a big fan of pilates, personally. It has it's merits, but it truly gets carried away when it comes to "stott" pilates.
Working out on inflatable items like a bosu ball, dyna disc, stability ball....are ways to make your environment more unstable, but unless you want to tryout for the circus ole....it's far from necessary.
I tell most people to get down basic movements. The core simply stabilizes and keeps the lumbar spine from becoming "mobile". Once the basics are understood start adding in more planes of motion (frontal, transverse)...begin to train three dimensional. Got that down, lets make things more unstable by removing some stability (balance on one foot, before hopping around on a inflaltable object)
Here are a few of my favorite "core" training movements
keep in mind, this is just my view