The theory starts from the premise that an organism's traits vary in a nondeterministic? way from parent to offspring, a process called "individuation" by Darwin. The theory of natural selection does not make any specific claims as to how this process works, although more recent scientific discoveries in genetics explain several mechanisms: sexual reproduction (which mixes the DNA of two parents into an offspring), random mutation, DNA transcription errors, and others.
If a particular variation makes the offspring which manifest it better suited to survival or to successful reproduction, that offspring and its descendents will be more likely to survive than those offspring without the variation. The original traits, as well as any maladaptive variations, will disappear as the offspring who carry them are replaced by their more successful relatives.
Therefore, certain traits are preserved due to the selective advantage they provide to their holders, allowing the individual to leave more offspring than individuals without the trait(s). Eventually, through many iterations of this process, organisms will develop more and more complex adaptive traits.
What makes one trait more likely to succeed is highly dependent on environmental factors, including the species' predators, food sources, physical environment, and so on. When members of a species become separated, such as geographically, they face different environments, and tend to develop in different directions. After a long period of time, their traits will have developed along different paths to such an extent that they can no longer interbreed, at which point they are considered separate species. This is why a species will sometimes separate into multiple species, rather than simply being replaced by a newer form of the species.
Additionally, some scientists have theorized that an adapation which serves to make the organism more adaptable in the future will also tend to supplant its competitors even though it provides no specific advantage in the near term. Descendants of that organism will be more varied and therefore more resistant to extinction due to environmental catastrophes and extinction events. This has been proposed as one reason for the rise of mammals.
Natural selection can be expressed as the following general algorithm:
Note that the above algorithm is made with no explicit reference to biological entities. Thus, a form of natural selection could occur in the non-biological realm (see, for example, Genetic programming). Note also that this formulation does not rule out selection occurring at all biological levels (e.g. gene, organism, group). Finally, note that the particular process of introducing new traits does not matter.
Perhaps the most radical claim of Darwin's theory of evolution through natural selection is that a set of random processes (including genetic mutation and natural disasters) can produce order. It is this fundamental claim that has inspired some of Darwin's most ardents supporters -- and that has provoked the most profound opporition. Some groups prefer to believe in divine intervention or guidance of the process; this is the "Intelligent design" school of thought.
Darwin first outlined his theory in two unpublished manuscripts written in 1842 and 1844 and more fully developed it for publication in The Origin of Species, especially Chapter 4.
See also: [Artificial selection]?, Sexual selection, Evolution
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