You probably want to have a look at an introductory course to evolutionary biology such as Understanding Evolution. In the meantime, here are some information that might help.
Darwin's evolution theory
The expression "Darwinian evolution theory" easily yield to missunderstanding because Darwin was probably the most important scientist (and one of the first if not the first) to develop evolution theory but not the only one. Evolution theory is not anymore Darwin's evolution theory.
Lewontin Recipe
The Lewontin recipe is a good way in order to understand what is natural selection and when it occurs. The Lewontin recipe says that natural selection occurs whenever:
- Individuals in a population varies in terms of a given trait
- This trait has some (additive) heritability. Here is one of the several posts that explain the concept of heritability. It might be slightly a post that is a bit advanced for you though but shortly speaking it means that offspring are more similar to their parents more than they are to other non-kin individuals in the population.
- The fitness varies (not necessarily linearly) as the trait varies.
Simple example:
- In a population, there are blue pens and red pens
- Reproduction is asexual and blue pens create other blue pens while red pens create other red pens.
- blue pens make more offspring than red pens.
In such situation natural selection occurs yielding the frequency of the blue pens to increase in the population while the frequency of red pens will decrease.
What is evolution?
Evolution is not only about natural selection. It is for example very important to consider random events. One of them is mutation, another is genetic drift (I am not trying to list every parameters that influence evolution but only to give you a sense of why natural selection is different than evolution with a goal of explaining why a trait that is needed do not necessarily appear). Both mutations and genetic drift explain why a species will not necessarily be perfectly adapted to its environment.
Mutations
In the broad sense mutation is any change in the DNA sequence. Some changes are more likely to happen that other but in any case the likeliness of these changes to happen is not dependent on the consequence they will have on the phenotype (shortly speaking, phenotype is how an individual looks like) and on the reproductive success. So mutations occur randomly and the specific mutation that would be needed in the population may not occur. Therefore saying, if a trait is needed (in the sense of "if a trait would be beneficial"), then a mutation will occur to make this trait existing is totally wrong. Note that most mutations are deleterious (decrease the reproductive success) while few of them are beneficial (increase the reproductive success) and those that are beneficial are more likely to raise in frequency in the population.
Genetic drift
If the change in frequency of mutations would depend exclusively on natural selection, then I would not have said before that a beneficial mutation is more likely to raise in frequency but I would have said that a beneficial mutation will raise in frequency deterministically. An intuitive explanation of what is genetic drift can be found on this post. It will also allow you to understand why small population undergo more random change in frequency of their genes than are big population.