In Short, Darwin's theory of evolution can be explained as follows
Increase in Population ( Overpopulation ) >>>> Struggle for existence >>>>> Survival of the Fittest
When population of an organism increase , competition for food , mating increases. This leads to the struggle for existence and the concept of survival of the fittest. Weaker one dies and the superior ones lives and reproduce.
infinite possible variation of organisms, natural selection of the best suited variations of the organisms, descent with modification resulting in increasing complexity due to natural selection.
Darwin's theory of evolution more correctly called descent with modification. postulates that all life started from a simple cell. ( Darwin later proposed that the first cell came from non living chemicals in a warm pond)
The overpopulation causes competition within the population. Darwin used Mathis observations as part of his theory. Since there were more members of the population than the environment could support only the best suited to the environment would survive.
This required an infinite possibility of variations within the population. Darwin didn't know what the source of the variation was he felt that all the possibilities were contained in the germ cells. Neo Darwinian evolution proposed that the source of variation was random accidental mutations.
Those organisms with the best suited variation would survive and pass those variations on to their offspring. The offspring would be more advanced. In turn the offspring would give rise to more advance offspring. This progression would result in descent with modification Darwin's theory of evolution.