typical contexts
A concrete category is a category that looks like a category of “sets with extra structure”, that is a category of structured sets.
A concrete category is a category $C$ equipped with a faithful functor
to the category Set. We say a category $C$ is concretizable if and only if it admits a faithful functor $U: C \to Set$.
Very often it is useful to consider the case where $U$ is representable by some object $c_0 \in C$, in that $U \simeq C(c_0,-)$. For example, this is important for the statement of various concrete dualities induced by dual adjunctions. We say in this case that $(C, U: C \to Set)$ is representably concrete. By definition, the object $c_0$ is then a generator of the category.
We remark that the existence of a left adjoint $F$ to $U: C \to Set$ implies that $U$ is representable by $F(1)$. Conversely, if $C$ has coproducts or even just copowers, then representability of $U$ implies that $U$ has a left adjoint.
One can also consider concrete categories over any base category $X$ instead of necessarily over $Set$. This is the approach taken in The Joy of Cats. Then the (small) categories concrete over $X$ form a 2-category $Cat(X)$.
The following furnish examples of concrete categories, with the first three representably concrete:
$C = Set$ itself with generator $c_0 = \{\bullet\}$ the singleton set.
$C = Top$ with the generator $c_0$ taken to be the one-point space.
Any monadic functor $U: C \to Set$ is faithful (because it preserves equalizers and reflects isomorphisms) and has a left adjoint. As special cases, we have the usual collection of examples of concrete categories: monoids, groups, rings, algebras, etc.
A category may be concretizable in more than one way:
Take $C$ to be the category of Banach spaces with morphisms those (everywhere-defined) linear transformations with norm bounded (above) by $1$ (so $\| T v \| \leq \| v \|$ for all $v$ in the source). Then there are two versions of $U$ that one may use: one where $U ( V )$ (for $V$ a Banach space) consists of every vector in $V$, and one where $U ( V )$ consists of those vectors bounded by $1$ (so the closed unit ball in $V$). The first may seem more obvious at first, but only the second is representable (by a $1$-dimensional Banach space).
Insofar as categories such as $Set$, $Top$, $Vect_k$, etc. admit many generators, these categories may be rendered representably concrete in a variety of ways. Indeed, the category $Vect_k$ may be monadic over $Set$ in many different ways. For example, if $V$ is $n$-dimensional, the functor $\hom(V, -): Vect_k \to Set$ is monadic and realizes $Vect_k$ as equivalent to the category of modules over the matrix algebra $\hom(V, V)$.
Any Grothendieck topos is concretizable, but not necessarily (and typically not) representably concretizable. If $E = Sh(C, J)$ is the category of sheaves on a small site $(C, J)$, we have a familiar string of faithful functors
But if for example $E$ is the category of sheaves over $\mathbb{R}$, then no object $X$ can serve as a single generator of $E$, since it cannot detect differences between arrows $Y \stackrel{\to}{\to} Z$ whenever the support of $Y$ is strictly contained in the support of $X$.
A concrete category that is equipped with the structure of a site in a compatible way is a concrete site. The category of concrete sheaves on a concrete site is concrete.
Every small category $C$ is concretizable (since it fully and faithfully embeds in the concrete category $Set^{C^{op}}$).
If $C$ is concretizable, so is $C^{op}$.
By assumption, there is a faithful functor $U^{op}: C^{op} \to Set^{op}$, and $\hom(-, \mathbf{2}): Set^{op} \to Set$ is monadic.
Of course, since a category $C$ may possess a generator but no cogenerator, it does not follow that $C^{op}$ is representably concrete if $C$ is.
A finitely complete category is concretizable, i.e., admits a faithful functor to $Set$, if and only if it is well-powered with respect to regular subobjects.
“Only if” was proven in (Isbell). To prove it, note that if $F: C\to D$ is a faithful functor, then it is injective on equivalence classes of regular subobjects. For suppose that $m\colon a \to x$ is the equalizer of $f,g\colon x\rightrightarrows y$, and $n\colon b\to x$ is the equalizer of $h,k\colon x\rightrightarrows z$. If $F(a) \cong F(b)$ as subobjects of $F(x)$, then since $f m = g m$ and so $F(f)\circ F(m) = F(g)\circ F(m)$, we must also have $F(f)\circ F(n) = F(g)\circ F(n)$; hence (since $F$ is faithful) $f n = g n$, so that $n$ factors through $m$. Conversely, $n$ factors through $m$, so we have $a\cong b$ as subobjects of $x$. Since $Set$ is regularly well-powered, it follows that any category admitting a faithful functor to $Set$ must also be so.
(Actually, Isbell proved a more general condition which applies to categories that may lack finite limits.)
“If” was proven in (Freyd). The argument is rather more involved, passing through additive categories, and is not reproduced here.
A relatively deep application of Isbell’s result is that the classical homotopy category Ho(Top) of topological spaces is not concretizable, even though it is a quotient of $Top$ which is concretizable. (Freyd 70)
A similar way to use Isbell’s result applies to show that a really vast number of model categories can not have a concrete localization at weak equivalences: see Di Liberti and Loregian, 2017
Peter Freyd, Concreteness, JPAA 3 (1973)
Peter Freyd, Homotopy is not concrete, in The Steenrod Algebra and its Applications, Springer Lecture Notes in Mathematics Vol. 168, Springer-Verlag, 1970, Republished in: Reprints in Theory and Applications of Categories, No. 6 (2004) pp 1-10 (web)
Ivan di Liberti, Fosco Loregian Homotopical algebra is not concrete, arXiv preprint.