Darwin's - Natural selection theory

Natural selection is the theory that particular traits of a given species are selected by how effectively individuals who have those traits pass those traits on to offspring. If a particular trait makes an individual more capable of surviving to breeding age and producing offspring, then that trait will slowly spread through the population as those individuals that have it breed and pass it on to their children. If a particular trait limits how well an individual can survive or breed, that individual will not produce as many children, and that trait will slowly disappear from the population because it is not being passed on.

The way the theory works is that first you have mutations: you might see traits X, Y and Z appear in a population simply because some gene was altered by some process. Usually scientists assume that gene mutations are random, but there is some debate on that issue. Once these new traits X, Y and Z exist, then natural selection comes into play. The traits that are most fit for the particular natural environment the population lives in tend to be passed on; the other traits tend not to be passed on.

For a silly example, consider the adaptation of M&Ms (if M&Ms were living creatures). So you have this bowl full of M&Ms of different colors: reds, greens, blues, and yellows. The colors were randomly determined originally, there's no real reason for them, and when the M&Ms produce children their children are the same color as one of the parent M&Ms (e.g. if a red and a blue M&M mate they will produce only red and blue children). So then you and I come along and start eating M&Ms, because yummm... and it just so happens that you like green M&Ms and I like red ones. So you eat the greens mostly and I eat the reds mostly, and in the bowl red and green M&Ms have a harder time surviving to mate and produce children, and so fewer and fewer red and green M&Ms are born, and so the bowl gradually fills with blue and yellow M&Ms. That's natural selection. Then let's say a new random mutation occurs: a purple M&M that tastes like dog poop. Neither one of us will eat that - we'd rather eat a blue or a yellow than those yucky purple ones - and so over time the bowl starts to fill with purple M&Ms, because they are the only ones assured of surviving our appetites. Again, a random mutation that leads to a new round of natural selection.

Natural selection is survival until reproduction, so it's kind of both. The more offspring you have, the better you have managed to pass the selection filter. (Note that there are all kinds of interesting details here, like organisms having a stake in the survival of the offspring of their close relatives because they share genes.)

Natural selection is, generally speaking, non-random. There are elements of randomness to what individuals survive, like if there's a lightning strike that kills one individual but not the one next to it, but unless those two individuals had different characteristics that made being struck by lightning more or less likely that event is not related to natural selection.

And in the long run, selection is clearly non-random. The events deciding whether a single individual, especially a short-lived one, will successfully reproduce will be somewhat random, but over many generations that randomness will accumulate into non-random patterns for which some individuals will be better suited than others.

Or you can look at it from the other perspective: What would the world be like if selection was completely random? That would mean that an individual's characteristics would have no effect on how well it survived and reproduced because those who survived and reproduced would be picked entirely at random. Clearly this is not the case, and therefor selection cannot be completely random.

Natural Selection is effectively the differential reproductive success of individuals.  Certain phenotypes have different fitnesses in a given environment, and natural selection is the process by which these fitnesses determine the "winners".  The "winners" will have better reproductive success, and their traits (phenotypes) will be preferentially transmitted to the next generation.

Perhaps to make an attempt at an analogy, think about natural selection like you choosing your favorite color.  This favorite color is the preferred trait, the trait that is being selected for, and you will select for it in, say, clothes.  Many of your clothing items will have this color, and if you ever decide that your favorite color is something different, your wardrobe will become more and more dominated by that favorite color, and you get the idea.

The problem with that analogy is that the trait that is selected for is not "chosen" per se.  You do not "choose" that a longer beak on a Galapagos finch will confer a feeding advantage.  Rather, it happens that in a specific environment (some of the Galapagos Islands, but not others) a longer beak will help the finch feed more easily, and thus live longer, having more chance to reproduce.

For the most part, traits that are naturally selected for confer a reproductive advantage in one way or another, even in species that do not sexually reproduce.

To rephrase all of this, Natural Selection is the differential reproductive success of individuals due to advantages conferred by a trait, and the increased occurrence of the trait over time (more individuals in later generations which have that trait).

Evidence for Natural selection theory -

A very simple, relevant example is occurring in the field of medicine today:
Bacterial drug resistance. 

Basically imagine a population of bacteria that cause harm to humans when infected. Now, an antibiotic drug comes along and wipes out nearly all of the population. However, a few of the bacteria from that original population survived due to some trait they possessed. Subsequent generations of those few surviving bacteria will select for that trait that makes them resistant to the antibiotic used. This is natural selection, favoring traits that increase fitness. 

On a larger scale, natural selection is a mechanism of evolution, since changes in allele frequencies become widespread within a population. In the case I gave about bacteria and antibiotics, this is known as the bottleneck effect, a form of genetic drift.

Source - Quara