How is "selection" best defined?
My favourite definition

Selection is a fitness differential associated to a genetic variance among individuals in a population.

You might want to have a look at What does fitness really mean?.
What I like about this definition
I personally went with this definition because it is short and clear cut and directly highlight the importance of mathematical statistics in the evolutionary biology literature.
What I don't like about this definition
This definition has the issue that it may be hard to be fully understood to a layman reader. Also, I think my definition lack a mention of the phenotype as being causal to the fitness differential. While such mention is not necessary, I think it would clarify in most minds what selection is really about.

Now, let's consider the definitions that we typically get when we google the question
Wikipedia
Here

Natural selection is the differential survival and reproduction of individuals due to differences in phenotype

I am quite unhappy with this definition. While I like that it highlights the fact that fitness difference is caused via phenotypic difference, it completely lack the idea that for a change in allele frequency to happen, this variation must be correlated to genetic variation.
I would note that the rest of the introduction on the wikipedia article clarify my criticisms. I only attack the short version (one sentence) version of the  definition.
Dictionnary.com
Here

the process by which forms of life having traits that better enable them to adapt to specific environmental pressures, as predators, changes in climate, or competition for food or mates, will tend to survive and reproduce in greater numbers than others of their kind, thus ensuring the perpetuation of those favorable traits in succeeding generations.

There are a few things I don't like about this definition
..traits that better enable them to adapt..
It sounds like they are talking about evolvability. I would just say traits that are better adapted.
..too much emphasis on the role of the environment..
Also, I don't like that it is presented as if all selection pressures where environmental (predators, climate, competition for food, etc...) and completely ignore a very important source of selection which are simply related to the functioning of the body. A mutation that causes an individual to fail copying DNA anymore will obviously be lethal and under very strong purifying selection. Such the &quot;outside-of-the-body environment&quot; is important but it is definitely not the only thing.
It is kind of a common misconception about evolution that selection occurs when there is a climate change and not so necessarily in other case in most people's mind.
..ensuring the perpetuation of those favorable traits in succeeding generations..
Well.. it does not &quot;ensure&quot; that these traits will be kept. First because it only increases the probability of certains allele to be kept or to reach fixation, it does not ensure it. Also, because depending on the genetic framework, once the underlying QTL have reached a given frequency, depending on the actual genetic mechanism (epistasis, dominance and others), traits that are being express might differ quite a bit from those that have been selected for.
Overall, I think the definition lack any notion to genetics.
Biology-online
Here

A process in nature in which organisms possessing certain genotypic characteristics that make them better adjusted to an environment tend to survive, reproduce, increase in number or frequency, and therefore, are able to transmit and perpetuate their essential genotypic qualities to succeeding generations

I like this definition. The only issues in my opinion are very similar to my above opinions

too much emphasis on the environment as done before
it seems that they define more the consequence of natural selection than what it actually really is.

What Google gave me when I entered 'what is natural selection?'

the process whereby organisms better adapted to their environment tend to survive and produce more offspring.

Again there is IMO, too much emphasis on the environment and no mention of thee actual mechanism (fitness - genetics correlation).

Potential confusion
Some people might use the term &quot;selection&quot; to refer to the existence of phenotype - fitness correlation and the &quot;response to selection&quot; as being the effect of such correlation of the population genetics whenever there is a non-zero association between phenotype and genotype. This nomenclature would typically be inspired by the terminology used in the Breeder's equation (see How to interpret the breeders equation?).
However, none of the definitions I could find clearly suggested that selection was nothing but a phenotype - fitness correlation as all of them implied that selection has a long term impact on the population phenotype which such phenotype - fitness correlation would not have in absence of a phenotype - genotype correlation.

Lewontin recipe
I am not sure a recipe count as a definition but Lewontin recipe is a good way to understand the what is natural selection

Individuals in a population varies in terms of a given trait
This trait has some heritability.
The fitness varies (not necessarily linearly) as the trait varies.

The recipe hides the complexities behind the term heritability. You can learn about the math of heritability at Why is a heritability coefficient not an index of how “genetic” something is? .