In his introduction to On the Origin of Species, Darwin gave a succinct summary of the theory:
In the next chapter the Struggle for Existence amongst all organic beings throughout the world, which inevitably follows from their high geometrical powers of increase, will be treated of. This is the doctrine of Malthus, applied to the whole animal and vegetable kingdoms. As many more individuals of each species are born than can possibly survive; and as, consequently, there is a frequently recurring struggle for existence, it follows that any being, if it vary however slightly in any manner profitable to itself, under the complex and sometimes varying conditions of life, will have a better chance of surviving, and thus be naturally selected. From the strong principle of inheritance, any selected variety will tend to propagate its new and modified form.
The fact that some heritable features give an organism a "better chance of surviving" in the "complex and sometimes varying conditions of life" (i.e., the conditions found in the organism's environment) can be taken as a definition of "natural selection". For example, if thicker fur gives some members of a population a better chance of survival in an arctic environment, and the result is that members of the population with thicker fur out-breed members with thinner fur, that would be an instance of natural selection in action. "Nature" simply refers to the sum total of all environmental conditions which might be responsible for giving different heritable features different chances of surviving and reproducing, nothing more is implied.
In modern mathematical models of Darwinian evolution this is formalized with the concept of different genomes having differential fitness, the probabilistic expected value for the number of offspring an organism with that genome would have in a specific environment (say, a modern arctic tundra). See "The propensity interpretation of fitness" in the SEP article on fitness, which says:
Among philosophers of biology there has been a wide consensus that the solution to the problem of defining individual ‘fitness’ is given by treating it as a probabilistic disposition. As such it causally intervenes between the relationship of environments to organisms that cause it, and the actual rates of reproduction that are its effects. Thus, fitness turns out to be a “garden-variety” dispositional concept like ‘magnetic’ or ‘fragile’. These properties, and all dispositions are distinguished both from the actual behaviour to which they give rise—e.g., attracting iron filing or breaking in the case of magnetism or fragility; some items are magnetic and others fragile without ever actually attracting iron filings or breaking. Similarly, an organism can have a probabilistic disposition to have n offspring and yet “unluckily” never actually reproduce.
edit: In the comments, @JD asks for evidence that when Darwin talks about the chances of survival "under the complex and sometimes varying conditions of life" he is referring specifically to chances given the organism's natural environment. To me this seems fairly self-evident, simply because I can't think of any other sensible interpretation of what the phrase "complex and sometimes varying conditions of life" could mean for a wild animal or plant (and the plural nature of the phrase doesn't seem to fit with the idea that he was talking about selection by a singular purposeful agent, whether 'God' or an anthropomorphized 'Nature'). But for further support for this reading, in chapter 3 he elaborates on the notion of natural selection:
Owing to this struggle for life, any variation, however slight and from whatever cause proceeding, if it be in any degree profitable to an individual of any species, in its infinitely complex relations to other organic beings and to external nature, will tend to the preservation of that individual, and will generally be inherited by its offspring. The offspring, also, will thus have a better chance of surviving, for, of the many individuals of any species which are periodically born, but a small number can survive. I have called this principle, by which each slight variation, if useful, is preserved, by the term of Natural Selection, in order to mark its relation to man’s power of selection.
Here he is referring to variations that give an organism a "better chance of surviving" in the context of its "infinitely complex relations to other organic beings and to external nature", which again sounds like odds of survival given the sum total of everything in its natural environment.
In chapter 4 he gives some examples of natural selection that again are all due to varied environmental factors, like a change in climate causing some species to go extinct:
We shall best understand the probable course of natural selection by taking the case of a country undergoing some physical change, for instance, of climate. The proportional numbers of its inhabitants would almost immediately undergo a change, and some species might become extinct.
Later in the chapter he gives some "imaginary illustrations" of natural selection:
In order to make it clear how, as I believe, natural selection acts, I must beg permission to give one or two imaginary illustrations. Let us take the case of a wolf, which preys on various animals, securing some by craft, some by strength, and some by fleetness; and let us suppose that the fleetest prey, a deer for instance, had from any change in the country increased in numbers, or that other prey had decreased in numbers, during that season of the year when the wolf was hardest pressed for food. Under such circumstances the swiftest and slimmest wolves have the best chance of surviving, and so be preserved or selected, provided always that they retained strength to master their prey at this or some other period of the year, when they were compelled to prey on other animals.
The deer are a part of the wolf's natural environment (specifically its 'relations to other organic beings' mentioned in chapter 3), and the "swiftest and slimmest" wolf therefore have the best chance of catching enough deer to survive.
Then there is this example:
Let us now take a more complex case. Certain plants excrete a sweet juice, apparently for the sake of eliminating something injurious from their sap: this is effected by glands at the base of the stipules in some Leguminosæ, and at the back of the leaf of the common laurel. This juice, though small in quantity, is greedily sought by insects. Let us now suppose a little sweet juice or nectar to be excreted by the inner bases of the petals of a flower. In this case insects in seeking the nectar would get dusted with pollen, and would certainly often transport the pollen from one flower to the stigma of another flower. The flowers of two distinct individuals of the same species would thus get crossed; and the act of crossing, we have good reason to believe (as will hereafter be more fully alluded to), would produce very vigorous seedlings, which consequently would have the best chance of flourishing and surviving. Some of these seedlings would probably inherit the nectar-excreting power. Those individual flowers which had the largest glands or nectaries, and which excreted most nectar, would be oftenest visited by insects, and would be oftenest crossed; and so in the long-run would gain the upper hand. Those flowers, also, which had their stamens and pistils placed, in relation to the size and habits of the particular insects which visited them, so as to favour in any degree the transportal of their pollen from flower to flower, would likewise be favoured or selected.
Again the interaction between the insects and the flower is just another environmental factor, one which favors flowers with stamens and pistils placed in such away that the insects are more likely to carry their pollen.