Refraction is simply the bending of light.  This happens when light travels from one medium to another.  In the case of Figure 5, the refracted ray is what goes through the surface of the water.  Notice that the incident ray does not go through the surface of the water in a straigth line.  In other words, the refracted ray is bent.  Therefore, the angle of incidence is not the same as the angle of refraction.

Different frequencies are bent in different amounts.  For example, since blue has a higher frequency, it is refracted more than red, which has a lower frequency.  The difference in the degree of refraction due to different frequency of light is called dispersion.  In the case of the rainbow, sunlight travels from the air in the atmosphere and enters into the raindrop.  It then travels from the raindrop back out to air.  Therefore, between the time when light is emitted from the sun to the moment when we see the rainbow, the light has actually undergone two refractions before it reaches our eyes!

Reflection and refraction do not always happen.  Sometimes they happen individually, but sometimes they happen together.  It all depends on the angle of incidence.