How does evolution occur?

The success of individuals in a population can be expressed in terms of fitness (chance of the individual and its offsprings to survive and to reproduce).
The theory suggests that the natural selection will favour traits that increase the fitness. As a matter of fact, and only if they are heritable, those traits are most likely to be passed to the next generation, and so on.
Since evolution acts on the genotype, it takes place on a long time-scale.

Example :
Take a random population. Let's say that the mean length of the individuals in this population is 10 cm. However, being big is advantageous in their actual environment. So, mainly the big ones survive and transmitt their alleles. On the next generation, since the alleles coding for a big length were mainly transmitted, the mean length in the population increased to 10,5 cm. Etc ... In the end, in this environment, the population will evolve toward bigger sizes.

Of course, this example is very simple and only consider one trait of the population. But, the same principle is applied to other heritable traits.

Quick recall : natural selection is the sum of all the environmental pressures (predation, weather, competition, ...) an individual can face in a given environment. It affects the organisms based on their phenotype.

Reminder : Evolution do not take place at the scale of the individual, but at the scale of the population.

Natural selection can only select from existing variations. These variation can create new types of the same species. Old forms of the species may become extinct. Natural selection can only cause extinction it can not create new forms.

Changes in the DNA must create new forms these changes do not occur by natural selection but by mutations. Mutations are thought to be random accidental changes in the DNA. These changes are always observed to be loses of information, not increases.

Evolution changes within a species occur by mutation. These changes make the species better adapted to the environment. Such as the movement of horses from from four toes to one toe. These changes however are not evidence of Darwinian evolution as the movement is from the more complex to the less complex.