When light transitions from one medium to another, it bends according to the refractive indices of the 2 media which it's transitioning between. However, different wavelengths of light bend differently because each wavelength experiences a different index in the material. When you shine light through a prism, the rainbow effect on the output is the result of this phenomenon. The visible wavelengths effectively refract at different rates and separate into their respective colors.

Dispersive power is basically a measure of the amount of difference in the refraction of the highest and lowest wavelengths that enter the prism. This is expressed in the angle between the 2 extreme wavelengths. The greater the dispersive power, the greater the angle between them, and vice-versa.