How does genetic variation affect the type of traits of adaptations in the individuals of a species?

Variation in and of itself does not affect the traits seen in the overall population. This is because gene frequency and subsequent changes in phenotype generally occur only when the phenotype is "selected" for or against in a population. Granted increased genetic variation is the potential for change, however just as in physics potential has to be acted on for change to occur.
The population of a species represents it's entire gene pool. The Hardy-Weinburg Principle states that the frequency of a gene remains the same as long as it is neither selected for or against.
Smaller populations can change rapidly due to genetic drift and isolation (which is more of a twist of random probability than selection, like getting heads ten times in a row tossing a penny) selection translates into change in a larger population.