Concrete category of algebras

Remarks

The functor $U$ is faithful because a homomorphism is determined by its underlying map.

Example: the category $\mathbf{Grp}$ of groups with its forgetful functor to $\mathbf{Set}$ is a concrete category of algebras.

Non-example: the category $\mathbf{Grp}$ with the functor that sends $G$ to the underlying set of its abelianization $G / [G, G]$ is not a concrete category, since this functor is not faithful. The two homomorphisms $\Z \to S_3$ that send $1$ to the transpositions $(1\,2)$ and $(1\,3)$ are different, but both induce the map $\Z \to \Z / 2\Z$ that sends $1$ to the nontrivial element.

Not every category admits a faithful functor to $\mathbf{Set}$: by a theorem of Freyd, the homotopy category of topological spaces does not.