Darwin's theory of Natural selection is usually described as "survival of the fittest." It is quite simple.
A generation of a certain animal may have a variety of traits. Different speeds, different color patterns, different beaks, different feet, different tail configurations, different lung capacities etc. Some of these traits may be more beneficial than others. Greater speed may make it easier to escape predators. Some color patterns may offer better camouflage. The more beneficial traits an animal has, the fitter it is to its environment, and the more likely it is to live longer and have more offspring. This results in future generations having a greater prevalence of the beneficial variations. Eventually these variations make one generation a distinct species from its ancestors.
Different environments can also separate two species from a single ancestor. In one environment one set of traits is beneficial, but in another those traits are a liability and another set is better (think country mouse and city mouse). This leads to future generations with more and more disparate traits until they become genetically incompatible resulting in two distinct species that are also distinct from a now extinct ancestor.