In the recent issue of the New Scientist, Richard Dawkins explains concisely what evolution is. It is very straightforward and clear.
The world is divided into things that look designed (like birds and airliners) and things that don't (rocks and mountains). Things that look designed are divided into those that really are designed (submarines and tin openers) and those that aren't (sharks and hedgehogs). The diagnostic of things that look (or are) designed is that their parts are assembled in ways that are statistically improbable in a functional direction. They do something well: for instance, fly.
Darwinian natural selection can produce an uncanny illusion of design. An engineer would be hard put to decide whether a bird or a plane was the more aerodynamically elegant.
So powerful is the illusion of design, it took humanity until the mid-19th century to realise that it is an illusion. In 1859, Charles Darwin announced one of the greatest ideas ever to occur to a human mind: cumulative evolution by natural selection. Living complexity is indeed orders of magnitude too improbable to have come about by chance. But only if we assume that all the luck has to come in one fell swoop. When cascades of small chance steps accumulate, you can reach prodigious heights of adaptive complexity. That cumulative build-up is evolution. Its guiding force is natural selection.
Every living creature has ancestors, but only a fraction have descendants. All inherit the genes of an unbroken sequence of successful ancestors, none of whom died young and none of whom failed to reproduce. Genes that program embryos to develop into adults who can successfully reproduce automatically survive in the gene pool, at the expense of genes that fail. This is natural selection at the gene level, and we notice its consequences at the organism level. There has to be an ultimate source of new genetic variation, and it is mutation. Copies of newly mutated genes are reshuffled through the gene pool by sexual reproduction, and selection removes them from the pool in a way that is non-random.
What makes for success in the business of life varies from species to species. Some swim, some walk, some fly, some climb, some root themselves into the soil and tilt green solar panels toward the sun. All this diversity stems from successive branchings, starting from a single bacterium-like ancestor, which lived between 3 and 4 billion years ago. Each branching event is called a speciation: a breeding population splits into two, and they go their separately evolving ways. Among sexually reproducing species, speciation is said to have occurred when the two gene pools have separated so far that they can no longer interbreed. Speciation begins by accident. When separation has reached the stage where there is no interbreeding even without a geographical barrier, we have the origin of a new species.
Natural selection is quintessentially non-random, yet it is lamentably often miscalled random. This one mistake underlies much of the sceptical backlash against evolution. Chance cannot explain life. Design is as bad an explanation as chance because it raises bigger questions than it answers. Evolution by natural selection is the only workable theory ever proposed that is capable of explaining life, and it does so brilliantly.
A huge problem with the public understanding of all things science revolves around terminology. Most people don't really know what a scientific "theory" is or "hypothesis." They typically prescribe the colloquial meaning, which is different. One of the things I'm working on with my elementary students is having them not only learn the correct definitions of these words, but then applying them to real experiments that they design (using a variety of plants). If we can correct these common misconceptions at an early age, kids will be that much more informed when they are adults.