The bent straw and the shallow pool
Put a straw in a glass of water and it looks snapped at the surface. A swimming pool always looks shallower than it is. A coin at the bottom of a mug hides behind the rim until you pour in water and it magically reappears. Every one of these is the same effect: refraction, the bending of light as it crosses from one transparent material into another. The straw is not bent — the light coming from its underwater part is, and your brain, assuming light travelled straight, places the straw where it isn't.
Why light bends: it changes speed
Light moves fastest in vacuum, at c. In glass or water it moves slower, because its oscillating fields keep being absorbed and re-emitted by the atoms. We package this slowdown into one number, the index of refraction n: the ratio of c to the speed v of light in the material. Vacuum has n=1 exactly; air is almost 1.00; water is 1.33; ordinary glass about 1.5; diamond a whopping 2.42. A bigger n means slower light.
The index of refraction is how many times slower light travels in a medium than in vacuum. Because v ≤ c, we always have n ≥ 1.
Snell's law: the quantitative rule
Here is the geometry. Picture a marching band crossing at an angle from a firm field onto muddy ground: the rank of players hits the mud one end first, that end slows, and the whole line pivots toward the mud. Light does the same. Going from a faster medium into a slower one (small n to large n), the ray bends toward the normal; going the other way it bends away. Snell's law makes this exact.
Snell's law. The product n·sin θ is conserved across the boundary; angles are measured from the normal. Bigger n on a side means a smaller angle on that side.
Total internal reflection
Now send light the hard way — from a slow, dense medium (large n_1) toward a fast one (small n_2), say from water up into air. The ray bends away from the normal, so as you increase the incidence angle the refracted ray tilts closer and closer to the surface. At one special critical angle \theta_c, the refracted ray would lie flat along the surface (\theta_2 = 90°). Push past it and there is no refracted ray at all — 100% of the light reflects back inside. This is total internal reflection.
Set θ2 = 90° in Snell's law to get the critical angle. It exists only going from denser to less dense (n1 > n2).
Total internal reflection is the physics behind optical fibres: light injected into a thin glass thread hits the walls beyond the critical angle and reflects perfectly, again and again, guiding the beam for kilometres with almost no loss. Your internet very likely arrives this way.
Dispersion: splitting white into a rainbow
One more twist: in glass, the index n depends slightly on wavelength — violet light is slowed a touch more than red. So when white light refracts, each colour bends by a different amount and the beam fans out into a spectrum. This is dispersion, and it is why a glass prism throws a rainbow and why raindrops paint one across the sky. A rainbow is dispersion plus one internal reflection inside each droplet, which is why the arc always sits opposite the Sun.