What is the opposite category of $Set$?

I only just saw this question. I'm a little surprised that none of the answers said concretely what $Set^{op}$ is. It's the category of complete atomic Boolean algebras, or equivalently complete Boolean algebras of the form $P(X)$ where $X$ is a set. The morphisms are morphisms of complete Boolean algebras (so functions that preserve arbitrary meets and negation, whence preserving the Boolean operations as well).

Note that a morphism of complete atomic Boolean algebras $\phi: P(X) \to P(Y)$ is uniquely determined by what it does to the atoms of $P(X)$. The atoms are of course singletons of $X$; if we denote $\phi(\{x\}) = S_x \subseteq Y$, then we can define a corresponding function $f: Y \to X$ to be the unique one such that $f^{-1}(x) = S_x$. Note that $S_x$ and $S_{x'}$ are disjoint if $x \neq x'$, because $\emptyset = \phi(\emptyset) = \phi(\{x\} \cap \{x'\}) = \phi(\{x\}) \cap \phi(\{x'\}) = S_x \cap S_{x'}$. This shows that $f$ is well-defined. This should give a clear idea why we obtain the opposite category.

The concrete manifestation of $X \to \emptyset$ in $Set^{op}$ is the unique map $P(X) \to P(\emptyset)$, where the codomain has just one element.

In abstract category theory morphisms are not required to be functions: we are given an arbitrary class, regarding its elements as arrows, equipped with domain and codomain information and an associative operation.

If we want, we can specify so that for a funtion $f:A\to B$, let ${\rm dom\,}f:=B$ and ${\rm cod\,}f:=A$ and the operation is defined as reversed composition of functions -- this is the category $Set^{op}$.
The same entity, $f$, as a function, plays a role of an arrow $f:A\to B$ in $Set$, but at the same time, itself is an arrow $B\to A$ in $Set^{op}$.

Anyway, the contravariant powerset functor establishes an equivalence between the opposite of category of finite sets and the category $BA_{fin}$ of finite Boolean algebras, and hence $$Set^{op} \simeq Pro(BA_{fin})$$ where $Pro$ denotes the pro-completion: i.e. taking all filtered limits formally.

This is a pretty old question, so I'm a bit suprised that nobody mentioned the most simple answer.

Since functions are the right-unique, left-total binary relations, they reverses are the left-unique, right-total ones. So, $\mathsf{Set}^\mathrm{op}$ is the category with sets as objects and left-unique, right-total binary relations as morhisms.

Regarding relations as subsets of the Cartesian product, the $X\to \emptyset$ morphism is the empty set. This is really a morphism in $\mathsf{Set}^\mathrm{op}$ because it is a left-unique and right total binary relation.

There is exactly one morphism from $X$ to $\emptyset$, namely the unique function from $\emptyset$ to $X$.