The easiest way to understand saturation is if you're familiar with the HSL (Hue, Saturation, Luminance) colour space. The easiest way to think of it is a colour wheel - the hue is how far around the wheel you are, Saturation is how far out from the center of the wheel, and luminance is how bright it is. Saturation adjustments push all colours further out from the center.

Vibrance is a "smarter" adjustment, that only pushes already intense colours (bright reds, blues, greens, etc.) more saturated, while leaving skin tones, and other normal colours alone. This tends to produce more natural looking images.

The easiest way to understand saturation is if you're familiar with the HSL (Hue, Saturation, Luminance) colour space. The easiest way to think of it is a colour wheel - the hue is how far around the wheel you are, Saturation is how far out from the center of the wheel, and luminance is how bright it is. Saturation adjustments push all colours further out from the center.

EDIT - My understanding of vibrance was incorrect. See the other answer. Leaving it here for anyone else who shares my preconceptions
Vibrance is a "smarter" adjustment, that only pushes already intense colours (bright reds, blues, greens, etc.) more saturated, while leaving skin tones, and other normal colours alone. This tends to produce more natural looking images.