What Evolution Is
Evolution is a scientific theory. You might be familiar with using "theory" in everyday contexts; typically it means a "guess" or a "hunch". In science, however, the word "theory" has a special meaning: it is a very specific explanation for a phenomenon that is backed by facts, observation, and repeated testing. There is a theory of gravity which explains how gravity works; likewise, there is a theory of evolution that explains how evolution happens.
In his 1859 publication of On the Origin of Species by Means of Natural Selection, Charles Darwin described the phenomenon of biological evolution as the descent of current life forms, by modification, from prior forms. To bolster the notion of common ancestry, he provided evidence from the structures of wild and domesticated plants and animals around the globe. Today, his findings have been extended by incorporating discoveries in the field of genetics, and we know more than ever about our species’ heritage.
The Theory of Evolution, which Darwin also described, explains how the current diversity of life on earth came to be. Evolution occurs because there is a finite amount of resources available, organisms produce too many offspring for all to survive, and variation is present in offspring. Because of these factors, a given population of organisms will adapt over time, by means of natural selection, to its environmental niche. Organisms that do not adapt will become extinct.
Evolution operates via the mechanisms of mutation, selection, genetic drift, and recombination. These mechanisms have been experimentally verified to produce diversity and adaptation, leading to the formation of new species in both the laboratory and in the wild.
For a well-organized and illustrated introduction to evolution we recommend the Evolution 101 tutorial. For a more precise definition of the term "evolution", see Laurence Moran's What is Evolution