For me a "model of (&infin;,n)-categories" is something (e.g., a model category) from which one can extract "the" (&infin;,1)-category of (&infin;,n)-categories.  One could make this more precise by choosing a preferred definition of (&infin;,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" (&infin;,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 <a href="http://mathoverflow.net/questions/8714/what-are-natural-transformations-in-1-categories/8959#8959">here</a> 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 (&infin;,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 (&infin;,2)-categorical analogues of the usual examples of 2-categories, for instance the (&infin;,2)-category of A<sub>&infin;</sub> ring spectra and bimodules, or the (&infin;,2)-category of (&infin;,1)-categories, or presentable (&infin;,1)-categories, or stable (&infin;,1)-categories, ...  For instance if you wanted to understand the relationship between (&infin;,1)-categories and their stabilizations&mdash;(&infin;,1)-categories form an (&infin;,2)-category, and stable ones form some kind of subcategory, and you might ask whether there is something like an adjoint to the inclusion&mdash;there isn't quite, but maybe there's an adjunction if we view these (&infin;,2)-categories as objects of some other (&infin;,3)-category.  So (&infin;,n)-categories do arise naturally in the study of (&infin;,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 (&infin;,n)-categories: Besides the iterated complete Segal space model we have Charles Rezk's &Theta;<sub>n</sub>-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&mdash;as far as I know none have been written down yet, except in the case n = 2.