All species show variation, some of which is caused by thier genes. Individuals with combinations of alleles that give them a better chance of survival than others with different combinations of alleles have a selective advantage. They are more likely to survive and breed, so thier alleles are more likely to be passed on to the next generation. The frequency of the avantageous alleles in the gene pool increases. This is known as natural selection.
Normally, when a species is already well adapted to its environment, and that environment is fairly stable, natural selection keeps things as they are. The frequencies of alleles in the population stay very much the same from generation to generation. This is known as stabilising selection.
If the environment and therefore selection pressures change, or if a new allele arises by mutation, then there may be a shift in the allele frequencies in subsequent generations. This is known a directional seection, and it can bring about evolution.
A population of organisms may become geographically isolated from the rest of the species. The selection pressures on this population may be different from those on the other populations of the species, and so the allele frequencies in the two groups begin to diverge.
Eventually, the allele frequencies may be so different in the new population that the individuals are unable to breed succesfully with the rest of the species, even if the geographical barrier is removed. A new species has been formed.