How is refraction related to dispersion?

You first think at refraction; a beam of light passes from a medium to another and get deflected. To distinguish these two media you use the index of refraction #n#, which is different for each medium.
But, the index of refraction #n=c/v# is basically telling us what is the velocity #v# of our light in that particular medium compared to the speed in vacuum #c#. The important thing is that the refractive index depends upon the wavelength!!!!

Now we get to dispersion!
A beam of white light passes from a medium to another BUT this beam is the composition of many chromatic components (colours each one at a certain wavelength) and each one has its "personal" index of refraction in the second medium!!!! So each component is deflected of a different amount (have a look at Snell's Law to "see" this deflection).

So basically light of different wavelength is bent of different amounts .

It is a little bit like to have a running competition with a bunch of runners grouped together. They reach a bend and after it they start to run on sand; each runner has a different "reaction" to the new surface (some run faster or slower depending on their velocity on the new surface) and after a while they "disperse" and are not grouped together anymore!!!

Hope it helps!