The “Theory of Evolution” was described by Darwin in 1859 in his famous book “On the Origin of Species by Means of Natural Selection”, subtitled “The Preservation of Favored Races in the Struggle for Life". There had been evolutionary ideas before Darwin’s time (such as the ancient “Great Chain of Being”) but Darwin was the first to popularize a reasonable scientific mechanism for how life forms could change through natural means.
“Neo-Darwinism” takes the ideas of Darwin and updates them with the science of “genetics” (instead of belief in the inheritance of acquired characteristics). Modern genetics developed after the time of Darwin.
The basic idea of organic evolution is that randomly occurring mutations introduce a small change into an organism. Natural selection retains those changes which end up providing the organism with some slight advantage, because the individual with the change is more likely to survive and reproduce at a higher rate, whereby the beneficial change is eventually spread throughout the entire population. “Differential reproduction” describes this idea.
The theory also speculates that all life, including both plants and animals, derived from a single simple life form through a very long series of small changes. “Descent with modification” describes this idea.
Mutation and natural selection CAN lead to the formation of a new species (populations that don’t normally inter-breed). Before Darwin most scientists believed that God created every species individually, but modern creation scientists now believe that the created “kind” was above the species level (probably approximately equivalent to the family, subfamily, or possibly genus taxonomic rank).