Darwin's Dangerous Idea, according to Daniel Dennett in the book by that name, is natural selection. This is often referred to as "Darwin's theory". But Darwin did not always think evolutionary events or processes were due to natural selection.
First off, let's say this again: Darwin did not think that evolution was due to chance. His mechanism of evolution was not "random chance", as Behe and others have asserted. He believed that variation was not correlated with fitness, yes, and occurred in ways that are statistically distributed (a conception not available until Galton came up with the notion some years after the Origin), but this is not "random chance" (sorry, Roberta*).
But Darwin also did not have a single mechanism of evolutionary change. While he often referred to the theory of natural selection as "my theory", or in correspondence with Wallace when Wallace proposed that natural selection could not account for the capacities of Mind, "your own and my child", he actually included a number of mechanisms as causes of evolutionary change.
It's best to see natural selection as the mechanism that explains, for Darwin, ecological adaptation and speciation. By adapting to novel conditions some varieties become more common in the population, and some do not then interbreed with other fit varieties, causing species. Let's leave speciation to one side for a moment and note that "natural selection" is the process of acquiring ecological fitness.
He also invented the notion of sexual selection. This explains traits that are not ecologically fit, but which contribute to reproductive success by triggering better chances of mating. Sexual selection explains ecologically unfit traits in one sex. The fitness is not ecological but is a matter of getting a mate (without which ecological fitness is evolutionarily irrelevant): it is reproductive fitness. Wallace, by the way, never accepted that female organisms could be responsible for sexual selection, although he allowed that some male mates might be.
Darwin also of course made analogy with the process of artificial selection. In this case the changes are due to an environmental factor - human intention. This is what we might call economic fitness, since breeders typically want varieties that serve economic purposes. The results, which as Darwin pointed out with pigeons, dogs, cattle and so on, are stunning, but the animals and plants often have problems due to a lack of sufficient time in which favourable variations (or, as we now call them, mutations) can occur to make up the shortfalls. For this reason, for example, pure bred dogs often have hip dysplasia, as selecting for one trait often leads to the unwanted inclusion of another. Which leads us to:
The "correlation of parts" - if you need to expand the length of the left leg, you are going to expand also the right. But this is trivial - Darwin expected that many more complex traits would change size and shape when attached to the development of a different part or trait that was undergoing selection. Today we call this, mainly in a genetic context, pleiotropy.
Then there is his "Lamarckian" mechanism of "use and disuse" - existing variants can be more strongly inherited because they are used. He needed this in part because it overcame some objections to the pace of natural selection being too slow to outrun the "regression to the mean" due to the blending of parental differences, and in part because it was received wisdom.
This is why Darwin wrote from the first edition of the Origin, that he thought that natural selection was the main, but not the exclusive, cause of evolution.
But this doesn't automatically license other mechanisms as being "Darwinian": for a start, orthogenesis, evolution that is directed from within towards a goal, is not Darwinian. Nor is saltative evolution, radical and full-formed changes. No mechanism in Darwin's writings permits that. If something - an organism, a trait or a populational property - is adaptive, it can only be via selection of one kind or another.
A nice essay by Gould on this is here.
* Usually critics of Darwin mean that there are no laws or reasons for the variations on which natural selection acts. This is not necessary, nor did Darwin think that this was true. But the variations are random in another sense, that they are not correlated with future needs of organisms, and hence fall into distribution curves.