Natural selection IS the mechanism. It is the mechanism for evolution. Populations are the unit of measure that evolves. After all, evolution is a change in the frequency of a trait over successive generations. An individual cannot change the frequency of its trait.
Fitness is the measure of success for an individual/trait in a particular environment. The individual with the most offspring will have more of its DNA in the next generation (assuming survivability) and so is the most "fit". Natural selection is not selecting "for" a trait. It is often better to remember that fitness is relative and the successful adaptation is not always the "best" in that environment. There are just too many random influences outside of the laboratory to assume that the fittest in one generation is necessarily the one that will, generation after generation, continue to be the fittest.
Natural selection basically poses that there is variability in organisms and variability in reproduction. Some of the variability in reproduction is due to the variability of the organisms. The interaction of the two is how the frequency of traits can change from one generation to the next.
To get to evolution, that variability has to work in favor of a trait in repeated, successive generations.