TITLE: A Characterization of Categories with a Conservative Forgetful Functor to SET
QUESTION [13 upvotes]: Examples of categories over $\bf{Set}$ such that the forgetful functor is conservative include the "algebraic" categories of groups, rings, modules, monoids, etc., but does not include the "higher-order" categories of topological spaces, smooth manifolds, schemes, etc. or some "relational" categories like posets.
Is there any intrinsic characterization of the difference in behavior here? Do we know that certain classes of categories always have conservative forgetful functors to $\bf{Set}$? 
I'd be particularly interested in a model-theoretic description (i.e. something like the fact that the first examples are categories of models of an algebraic first order theory while the later ones are not), but I'd also be interested in a description in terms of categorical properties.
EDIT: As pointed out in the answers, it's often possible to give a conservative faithful functor to $\bf{Set}$. However, these aren't the "underlying set" forgetful functors; they include extra information. Maybe the question becomes more interesting if we require the functor to be "canonical". This doesn't make much sense in terms of categories alone, but perhaps it does in terms of model theory: given a (perhaps higher-order) theory $\mathbb{T}$, when is the forgetful functor $\text{MOD-}\mathbb{T}(\bf{Set}) \rightarrow \bf{Set}$ conservative?

REPLY [6 votes]: Freyd proved in "On the concreteness of certain categories" that if $C$ is locally small than there is a conservative functor $C \to \text{Set}$.