The theory of evolution is generally referred to as 'Darwin's theory of evolution' because he was the first to offer an explanation of it in his book 'The Origin of Species' (1859). Others, such as the Greek philosopher Anaximander, had speculated centuries before that species evolved, and by the 19th century most scientists studying the fossil record realised that species changed over time, but Darwin was the first to propose the mechanism through which evolution occurred: natural selection. Later scientists have incorporated knowledge about genetics and genetic mutations into the theory.
'Biological evolution, simply put, is descent with modification' . In summary the theory of evolution
by natural selection is that random genetic mutations occur naturally within all species. Some mutations have beneficial effects, some have no effect and others have detrimental effects. Individuals with beneficial mutations tend to have a higher survival rate than the others, and so are more likely to reproduce and pass on the beneficial genes.
Over a long period of time, the beneficial genes accumulate, but different genes are beneficial in different ecological niches. In adapting to different (or changing) environments over many generations very small changes add up to large changes, and populations eventually become recognisably different species, and different species to their common ancestor.
Evolution is essentially nature's form of selective breeding. In only a few hundred years humans have managed to produce dramatically different breeds of all domestic animals, by favouring certain traits over others and allowing only those with the desired traits to reproduce. The Chihuahua and the Great Dane have the same ancestor (the wolf), for example.
Natural selection works over millions of years rather than a few centuries, and there has been life on Earth for over 3,500 million years . With such dramatic changes produced in only centuries by selective breeding, it is easy to see that over millions of years natural selection would produce recognisably different species. Even tiny differences in genetic material produce entirely different species. Only 1.5% of the DNA in chimpanzees and humans is different .
There is a common misconception that the theory of evolution suggests humans evolved from apes and that there is a missing link between them. This is simple (but often deliberate) misunderstanding. The theory of natural selection actually postulates that apes, other primates and humans all evolved from the same ancestor. The genetic code of the DNA in chimpanzees is 98.5% identical to that in humans , and it is therefore impossible to imagine that we are not related.
The theory of evolution postulates that all living things on Earth had the same ancestors: simple organisms that evolved in the 'primordial soup' billions of years ago. Scientists have shown  that in conditions similar to those believed to have existed around the time life arose, quite complex amino acids and the building blocks of DNA arise spontaneously even in one week. Evolution had over a billion years to work in.
There are several theories of how the first organisms arose, but there does exist today an intermediate between life and non-life: the virus.
 Berkely University: http://evolution.berkeley.edu/
 Fossil mall: http://www.fossilmall.com/Science/GeologicalTime.htm
 ABC 27 May 2004: http://www.abc.net.au/science/news/enviro/EnviroRepublish_1117169.htm
Charles Darwin, 'The Origin of Species' 1859, reproduced in the Harvard Classics series, Grolier,1981.