Yes, so:

A <a href="http://ncatlab.org/nlab/show/simplicial+set">simplicial set</a> is indeed precisely a <a href="http://ncatlab.org/nlab/show/presheaf">presheaf</a> on the <a href="http://ncatlab.org/nlab/show/simplex+category">simplex category</a>.

There are various <a href="http://ncatlab.org/nlab/show/model+category">model category</a> structures on categories of presheaves in general and on simplicial sets in particular.

With respect to the <a href="http://ncatlab.org/nlab/show/model+structure+on+simplicial+sets">standard model structure on simplicial sets</a> the <a href="http://ncatlab.org/nlab/show/Kan+complex">Kan complex</a>es are precisely the fibrant objects.

With respect to the local <a href="http://ncatlab.org/nlab/show/category+of+sheaves">model structure on presheaves</a> on a <a href="http://ncatlab.org/nlab/show/site">site</a> the <a href="http://ncatlab.org/nlab/show/sheaf">sheaves</a> are precisely the fibrant objects.

There is a very useful combination of these two statements:

A <a href="http://ncatlab.org/nlab/show/simplicial+presheaf">simplicial presheaf</a> is a presheaf on the product category of the simplex category and some site.

in the local projective <a href="http://ncatlab.org/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> the fibrant objects are precisely those simplicial presheaves that are Kan-complexes over each object of the site and that satisfy the  <a href="http://ncatlab.org/nlab/show/descent">oo-version of the sheaf condition ("descent")</a>: these are the <a href="http://ncatlab.org/nlab/show/infinity-stack">oo-stacks</a> on the given site.