Shankara is often labeled as an idealist. This is more than slightly puzzling, since he is famous for arguing against the idealism of Buddhists. My suspicion is that this is a byproduct of shift in meanings. If you look at the history of calling Shankara an idealist, it goes back to the nineteenth century; but 'idealism' has a lot of meanings in the nineteenth century, and even into the twentieth. I'm reminded of Norman Kemp Smith's influential discussions of whether Hume is a Naturalist or a Skeptic. What is often forgotten is that these two are part of a triad, which Kemp Smith also uses elsewhere, and 'Idealism' is the third branch (which Kemp Smith, of course, takes Hume clearly to oppose). But 'Idealism' in this sense means more or less the metaphysical view that mind or minds are fundamental to reality; almost all theists count as Idealists in this sense, for instance, and, indeed, Shankara is undeniably a kind of Idealist in that sense, if that's your definition of the word. Over time, however, the term has narrowed, but labels for philosophical positions have a certain sort of inertia, and change only slowly.
There are things that Shankara and other Advaitins say that can be suggestive of idealism; for instance, the claim that things are related to Brahman as an illusion is to the magician sustaining it. But Shankara uses a lot of analogies to describe the relation between the world and Brahman, and a certain art is required to navigate them. For instance he uses material analogies (e.g., the world is to Brahman as shape of a ring of gold to the gold of the ring) when discussing Brahman as principle of origination, but he is also very clear that Brahman itself is changeless and has no modifications; a naive treatment of the analogy without the negations would have Brahman changing and modified. What the rejection of change and modification does is show that the analogy is picking out not modification of a changeable substance, but some other aspect of the shape-gold relation -- Brahman is a reason for the coming-to-be of the world that has some features like those we find when we consider gold as a reason for the coming-to-be of shape, but this does not mean that the world is the shape of Brahman-stuff. When talking about Brahman as principle of subsistence, he often uses non-material analogies, like the illusion-magician one. But Shankara is again very clear that external objects really exist -- as he concisely puts it, and defends at length, the nonexistence of external things cannot be maintained because we are conscious of them. The point is not that the world is fictional but that it only subsists wholly in the action, if we may so speak, of Brahman, as an illusion only subsists in the action of the magician. It's the same with dream analogies: the claim is not about the nature of the world or the nature of Brahman, but about the nature of the relation between them, as the dream exists wholly and in every way because of the dreamer, so the world, despite not being a dream, exists wholly and in every way because of Brahman: this is possible wholly because that is actual. And these kinds of analogies are in fact not very different from analogies that are often used by Neoplatonism (reflection, shadow, participation=parceling-out), which is certainly not an idealism in the more modern and narrower sense.