Hydraulics are not only stronger than cable discs, but they don't need the frequent adjustments cables do and they make for much nicer feel and feathering, and you gain more stopping power with less effort. They have become very reliable, and even better, have come down in price, and in a few years (mark my words) cable discs will be history. My last bike had cable discs and the hydraulics on my new bike are superior in every way. Hydraulics are pretty much full proof, and while bleeding is a pain, should only need doing rarely.
Then there is V-brakes, which are the cable pull brakes mounted over the wheel. The advantages are that they are very strong (think of the rim as a massive 26" rotor), cheap, light, and easy to work with. The only real negative is that they don't respond well in wet or muddy conditions, and they are prone to overheating under major use (like downhill trail runs).
With cable discs, the cable pulls just like with regular cantilever brakes, except the cables pull the brake pad against the rotor by use of a caliper to create friction and cause slow down.
With hydraulics it's the same idea, except the fluid is used to activate the calipers. We know from grade 4 science class that liquid can't be compressed, and this is what makes hydraulics so effective.
Entry level brakes, particularly cable discs, have only one caliper - so on one side the pad is fixed and it's only pushed from the other side. Better brakes have dual calipers, which grip from both sides.