Charles Darwin (1809-1882) is famous for developing a way for evolution to occur, natural selection. It should be pointed out that when Darwin was alive there was no question that species evolved. Scientists in general all agreed that the species that were alive around them did not remain fixed forever and ever, but rather changed over the course of long spans of time. The problem was that no one could figure out how this took place. Darwin made observations of the natural world and noticed four simple features that would result in species changing in response to the natural environment. The process of change that Darwin proposed occurs as an inevitable consequence of these four conditions, and does not require any divine influence. The four conditions that Darwin elucidated were variation, heritability, superfecundity, and non-random mortality.
Variation means that that each individual in a population is unique. These differences may be very minor, but they are always there. This is so obvious a fact that it almost does not need to be spelled out. You are a unique individual who has never occurred before and will never occur again, and the same is true of every other species.
Heritability means that each individual will tend to pass on the variations it has to its offspring. In this way, the variations that are present in a population will tend to be passed down through the generations. In other words, short individuals will tend to have short offspring, etc. This heritability is not perfect, in most traits, because there is mixing between the traits of each parent.
Superfecundity means that more young are produced than can possibly survive. Each individual strives to pass its genes on into the future. To accomplish this, the more offspring produced the better and since all organisms use this strategy, a great many offspring are produced. This leads to competition among unique individuals for a limited number of available resources needed for life.
Non-random Mortality means that how dies matters. Many organisms die, and this is especially true of young organisms. This is driven,in part, by the competition mentioned above and, in part, by factors such as harsh weather conditions and other environmental factors. But while it is a foregone conclusion of superfecundity that some individuals will die, it is the fact that these deaths do not occur at random that allows for population-level changes to occur. In other words, the survivors survive for a reason, and the reason is that they have some advantage, however small, over those that did not survive. The survivors are then able to pass their advantages, whatever they are, on to the next generation.
After generations and generations of this combination of conditions, populations of individuals become evermore adapted to the environments in which they live, and so evolution by means of natural selection occurs.