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I just want to understand the concept of natural selection and its relation to evolution.

Evolution by natural selection occurs when we have hereditary trait(s) that causes an effect on reproduction rate of a population, either towards the positive, and thus deemed as beneficial or towards the negative and thus labeled as harmful.

Evolution by natural selection would result in keeping the beneficial trait(s), and wiping out the harmful ones.

The idea according to Darwin is that the slightest harmful effect a hereditary trait(s) would bring to a population then with time it would result in extinction of the population with that trait(s). [On the origin of Species, p.78, chapter: natural selection: "On the other hand we may feel sure that any variation in the least degree injurious would be rigidly destroyed"].

Therefore natural selection could be viewed to result ultimately in washing out of injurious traits, and only promoting beneficial traits. So natural selection can be the cause of extinction of some genotypes, and ultimately of some species.

In this sense natural selection can be considered as a creative natural tool, like a rubber wiping bad intermediate solutions, and only keeping the correct steps, thereby facilitating reaching into a solution.

Now suppose that we have a population of living subjects with varying heritable traits on an Island, and those had equal fitness over a very long period of time, so they were pretty much adaptive to their environment. Now suppose a volcano erupted or a big comet stroked that island that ended all life in it. Lets suppose that the genotypes of that population is not present outside of that Island. So these genotypes became extinct.

Here this event doesn't have the same functional genre that natural selection had, it actually didn't act in a differential manner promoting those genotypes with better fitness and wiping out those with lower fitness. No it actually was NOT fitness dependent at all, it simply squashed all life forms without regards to whatever feature was brought up by the different genotypes, all were simply extinguished.

Now can we label this event of indiscriminative extinction to be the work of natural selection also? Or we just label it as inadvertent environmental factor?

If the latter, then are those inadvertent environmental factors part of the mechanism for evolution?

The idea according to Darwin is that the slightest harmful effect a
hereditary trait(s) would bring to a population then with time it
would result in extinction of the population with that trait(s).

Citation? I certainly don't think any modern biologist would claim that a single deleterious allele will doom a population.

Therefore natural selection could be viewed to result ultimately in
washing out of injurious traits,

In real life, there is no infinity far away "ultimate" point to be reached. What's deleterious today might be beneficial if the environment changes. And the environment is always changing.

Now can we label this event of indiscriminative extinction to be the
work of natural selection also?

I don't see why you would, since survival was not influenced by genetics.

If the latter, then are those inadvertent environmental factors part
of the mechanism for evolution?

Well, if it kills every single organism, I don't think we can say that population is evolving any more. But bottlenecks randomly and radically changing allele frequencies are hardly an unknown mechanism for evolution.

Natural selection is an important part of evolution, but not the only part

Evolution is described as changes in heritable characteristics over time: you can look at changes in allele frequencies and call that evolution.

Genetic drift, for example, describes genetic changes that are caused by random sampling rather than selection. A natural disaster (if we assume it acts only to kill off some individuals, rather than changing the environment), could hypothetically contribute to genetic drift.

Consider some population of a species of birds on a group of islands. The birds can fly between islands, so they interbreed but not at the same rate (i.e., they are more likely to breed within their island than between islands). Therefore, if we look at allele frequencies, they are not identical. They will be especially likely to drift apart if they aren't especially important to survival (an example in humans could be something like eye color).

One day, all the birds on one of the islands die by some disaster. If we look at allele frequencies across the whole population across all the islands, those frequencies just suddenly changed: any alleles that were more common in the doomed island were just reduced. Evolution just happened (heritable characteristics just changed), with no selection.

These effectively random changes can be quite important evolutionary forces over the long run, especially when population bottlenecks occur, in which case a small population survives rather than is wiped out.

The catastrophe itself is an example of a change in environment that brings with it a change in selective pressure. The events you describe (a volcano eruption, or a comet strike) would select for existing variation in the population. Heritable traits that are more common in those surviving the event would have been selected for. This represents natural selection.