For me a "model of (∞,n)-categories" is something (e.g., a model category) from which one can extract "the" (∞,1)-category of (∞,n)-categories. One could make this more precise by choosing a preferred definition of (∞,n)-categories and asking for things equivalent to it. Of course it's currently less clear than for, say, models of spaces or spectra that there really is a unique "correct" (∞,1)-category which really is equivalent to everything we hope it's equivalent to. And indeed already for n = 1 there are related but distinct useful notions of category as I write here which manifest themselves in homotopy theory as Segal spaces and complete Segal spaces. But I think most everyone expects that there's a natural notion of (∞,n)-category up to n-categorical equivalence which is the right analogue of n-category (whatever that means).
The easiest examples are of course in the case n = 2, where we have (∞,2)-categorical analogues of the usual examples of 2-categories, for instance the (∞,2)-category of A∞ ring spectra and bimodules, or the (∞,2)-category of (∞,1)-categories, or presentable (∞,1)-categories, or stable (∞,1)-categories, ... For instance if you wanted to understand the relationship between (∞,1)-categories and their stabilizations—(∞,1)-categories form an (∞,2)-category, and stable ones form some kind of subcategory, and you might ask whether there is something like an adjoint to the inclusion—there isn't quite, but maybe there's an adjunction if we view these (∞,2)-categories as objects of some other (∞,3)-category. So (∞,n)-categories do arise naturally in the study of (∞,k)-categories for k < n. These examples are in a sense "algebraic" objects, as opposed to bordism categories, which turn out to be algebraic too, in a sense, but a priori are given by geometric constructions.
As for models for (∞,n)-categories: Besides the iterated complete Segal space model we have Charles Rezk's Θn-spaces, and I think simplicial strict n-categories are also supposed to give the right notion. There's the "complicial sets" model, which seems to me to be more conjectural. I would also like to hear about results about equivalence of these models—as far as I know none have been written down yet, except in the case n = 2.