Whoever recommended Dawkins to you, don't listen. Don't support Dawkins. He's an anti-religious fanatic who needs a sanity check.
Anyway, those are my 2 cents on Dawkins, on to evolution. I guess I could summarize it as such: natural selection and mutation. Animals are under constant threat, whether by disease or other animals or random resource deprivation (drought, famine, etc.) or, frankly, dumb luck (you tripped over a rock, broke your leg in half, and died due to infection). Animals who are more fit to withstand this, or have some natural talent or advantage, will be less likely to die young than others.
In theory, the general pattern of natural selection is that animals whose genes allow them to live to reproductive age and produce viable offspring will pass their genes to the next generation, thus prolonging certain traits. If Gazelle A and B are almost identical but A has a gene that makes it run twice as fast as B, A will have a much easier time getting away from predators. This is a basic, oversimplified model, but yeah. So Gazelle B gets eaten by lions, and A gets to spread his genes, and now the new generation has a higher chance of picking up A's fast-running gene.
Genes are constantly mutating, and while most mutations are bad and result in the cell instantly dying due to faulty processes in the cell, a few of them actually do good things. Through enough iterations of these mutations, some sperm or egg cell will have mutations that allow the offspring to develop a variant on an old trait. This variant, if good, will allow better survival in the current environment and that animal will go on to mate a lot and make lots of superior babies. If it isn't, the animal will lag behind and get eaten, hurt, whatever, and die.
This is, in theory, how it works. Of course, the superior animal could get hit by lightning and die before sexual maturity. Sure. But over millions of years, this pattern averages out to a set of creatures that constantly changes to better suit their environment.
There are all sorts of sub-topics here, like how genes actually change and how sex cell creation and fertilization work, what animals evolved from what, ecosystems, human-induced evolution (did you know some moths have changed color to suit human environments?), how different species split off, all that.