There are, due to genetic variation, always differences between individuals within a population. Everyone in a given room has countless differences with everyone else: different hair colors, heights, weights, metabolisms, etc.
While not all differences can be directly blamed on genetic variation, nearly all are at least influenced by it (and many are directly caused by it).
Due to these differences between individuals in a given population, each individual has a different biological fitness, or ability to survive and reproduce.
It is tempting to think of fitness as purely physical fitness, but keep in mind that the ultimate measure of fitness is passing on your genes to the next generation. As a result, even if you are the least physically-fit elephant in the herd, you might have the most biological fitness if your offspring do very well.
As described by Darwin, natural selection is simply the process by which less biologically-fit individuals have fewer successful offspring than individuals who are more biologically fit. Recall that an organism's physical traits and behaviors constitute its phenotype. The genetic components that contribute to those traits make up its genotypes.
For some insight into the process, read the following quotation from Alfred Russel Wallace regarding the theory he independently came upon:
"Vaguely thinking over the enormous and constant destruction which this implied, it occurred to me to ask the question, Why do some die and some live? And the answer was clearly, that on the whole the best fitted live. From the effects of disease the most healthy escaped; from enemies, the strongest, the swiftest, or the most cunning; from famine, the best hunters or those with the best digestion; and so on."
Natural selection acts upon an individual's phenotype, but its phenotype is caused by its genotype, at least partly. As a result, we can measure the impact of natural selection by measuring phenotypes or genotypes over time within a given population. To understand the interplay between these two in the process of natural selection, read the following excerpt from Dr. Mukherjee's excellent book The Gene: An Intimate History:
"The process of selection, notably, acts on a physical or biological attribute—and the underlying genes are selected passively as a result.
...
Phenotype, in short, drags genotypes behind it, like a cart pulling a horse. It is the perennial conundrum of natural selection that it seeks one thing (fitness) and accidentally finds another (genes that produce fitness). Genes that produce fitness become gradually overrepresented in populations through the selection of phenotypes, thereby allowing organisms to become more and more adapted to their environments. There is no such thing as perfection, only the relentless, thirsty matching of an organism to its environment. That is the engine that drives evolution."
- The Gene: An Intimate History by Siddhartha Mukherjee
Over time, then, positive traits will spread throughout a population across multiple generations. These positive traits that increase an organism's biological fitness are referred to as adaptations.
In order to illustrate natural selection at work, consider the peppered moths (Biston betularia) shown camouflaged here. These moths are a famous illustrative example of natural selection at work in recent history.
This population of moths, located in Manchester, England, was studied by scientist R.S. Edleston in the 1800s. In 1848, he found an unusual dark variant of the moth, the first recorded sighting of this. Overall, however, the population was still mostly the classic light-colored variation.
Over the subsequent half century, however, the peppered moth population underwent revolutionary change. By 1900, the peppered moth populations were found to be mostly dark (up to 98% dark to 2% light in some areas).
Clearly, we can now see that this is evolution in action - a population is changing over time and natural selection is acting on the variation of the population and their differential biological fitnesses.
It is important to clarify that these changes were found to be due to genetic changes in the population. In other words, it's not that their colors were changed due to someone painting all the moths black, for instance.
A change in populations due to natural selection can be traced back to the selection pressures to which the populations are exposed. Up until 1848, clearly the most common variant of the moth was the light coloration. However, if the environment changes, so, too, can the selection pressures.
What is beneficial in one environment can be deleterious in another. Thus, an adaptation is always context-dependent. No trait will be beneficial in every single circumstance.
For instance, a polar bear's thick coat is excellent for cold weather. But if temperatures start to increase (as they have), that heavy coat will quickly become a negative thing.
In 1896 (by now, people knew about natural selection), J. W. Tutt had an explanation to propose. This change to the environment was due to the growing Industrial Revolution affecting England heavily at the time. The smoke and pollutants from nearby industry had altered the color of the forests of England. He suggested that the camouflage of the light moth no longer functions in the darkened forest. As a result, dark moths lived longer and bred more than their lighter counterparts.
This, like all evolution, is an ongoing process. In the last fifty years, most industrial countries have taken measures to reduce their pollution. As a result, forests are remaining lighter and the number of dark moths are dropping in comparison to the lighter morphs. So evolution can actually reverse itself if conditions reverse (and if there is still variation in the population.
With a large enough sample size (or number of organisms), populations tend to show normality, or a normal curve, for most traits.
Essentially, the average individual tends to have a trait value that is toward the middle, but there will be small numbers of individuals with traits far above and some far below that average.
Normal curves are very common when studying traits at the population level. Therefore, it is important that we can analyze how these normal curves may change with evolution.
Variants at one end of the curve are favored, selecting for movement of the average to the left or right. Average changes.
Example: Dark-colored peppered moths are better camouflaged against a sooty environment, keading to more individuals in the population on the dark side of the curve.
Variants near the average (center) of the curve are favored, selecting for more individuals near average. Average stays the same.
Example: Robin nests contain, on average, 4 eggs. Larger clutches may result in malnourished offspring while smaller clutches may result in no viable offspring.
Variants at both ends of the curve are favored, leading to fewer individuals near the average. Average stays the same.
Example: Grey and mixed rabbits are better able to blend into a rocky environment than white rabbits. As a result, populations tends toward either end of the curve.
In order for natural selection to occur, there are certain requirements that must be met by the population. Essentially this is a process that requires materials to occur, and those materials must exist in the population.
As established, populations have differences amongst their individuals. This is the result, primarily, of genetic mutation. Genetic mutation, generally caused by mistakes in cellular processes, creates individuals that have unique combinations of DNA even in asexual organisms such as bacteria.
Sometimes these differences are obvious, sometimes they are subtle or behavioral.
"To Darwin, if the motor for evoluton is natural selection, then variation among individuals is its fuel. If individuals in a species vary in having traits that look and function differently, and some of those trains enhance the success of those individuals in a particular environment, then over time those creatures and traits should increase. If a trait is harmful, then it will diminish over time. The essence of evolution is variation among individuals. If all individuals in a population are exactly alike, evolution by natural selection could never happen."
- Some Assembly Required: Decoding Four Billion Years of Life, from Ancient Fossils to DNA by Neil Shubin
If all individuals were the same, there would be no measurable differences over time within a population. You would not be able to notice a trait become more or less common over time.
That variation must be able to be passed down from generation to generation in order to allow natural selection to act upon it.
All traits that are genetically-based fall under this category. However, some behaviors are learned and passed down across generations by parents to offspring. Natural selection can act on these traits as well.
Resources are finite, of course, regardless of the ecosystem. Eventually, resources will become scarce if a population continues to grow.
As a result, individuals will have to compete with one another to get resources, whether those resources be access to a mate, food, shelter, etc.
Natural selection acts on the biological fitness of individuals within a population. If all individuals have the same fitness, all individuals would have the same chances of surviving and reproducing.
This means that natural selection will not 'select' for any particular individual over any other. Thus, the only changes that may occur within the population would be due to chance alone, not selection.
So for natural selection to occur, some individuals have to be better competitors than others.