You'll find dozens of brushes out there, in all shapes, sizes and textures. If you're big into eye makeup, you may very well want to use them all. Most of us, though, will only use a few:
This one is small, with stiffer bristles, to allow for precision in smoothing and blending concealer over under-eye discolorations (and other blemishes). Use this one in firm, gentle strokes.
Eye Shadow Brush
The applicator included with your eye shadow isn't so great at applying eye shadow. It can leave streaks, and it's hard to clean. (It's not bad at smudging eye liner, though, so you don't throw it out.)
Go for the small (perhaps 1/2-inch), slightly rounded brush with medium-firm bristles, and try not to press too hard or you'll get shadow everywhere. Wash this after each use.
Unruly eyebrows can ruin the look of an otherwise perfectly made-up face. An eyebrow brush is small and hard; its bristles will smooth any stray hairs into place.
As for whether your bristles should be natural or synthetic, for most of us non-pros it doesn't matter all that much. The big consideration is allergies: If you have very sensitive skin, you may want to play it safe and go synthetic.
Brushes are definitely the bulk of your toolkit, but to round it out, you'll probably need two more pieces of equipment.