The question is elementary and was already answered in the comments. I'm posting a cw answer so that the question can be ticked as answer (otherwise it remains regularly bumped).
Let $I_D$ be the set of irreducible elements, $I_D/D^\times$ its quotient modulo multiplication by invertible elements, and $\mathcal{P}(X)$ the power set of $X$.
Then the map $S\mapsto D[S^{-1}]$, from $\mathcal{P}(I_D)$ to the set $\mathcal{L}_D$ of subrings of $\mathrm{Frac}(D)$ containing $D$, is surjective, and induces a bijection from $\mathcal{P}(I_D/D^\times)$ to $\mathcal{L}_D$.
The surjectivity follows from the fact that if $R\in\mathcal{L}_D$ and $a/b\in R$ with $a,b$ coprime, and $s$ is irreducible and divides $b$, then $1/s\in R$. This is an easy consequence of Bézout.
The injectivity is easy: clearly the set of elements of $D$ that is invertible in $R$ is invariant by multiplication by $D^\times$, and if $S$ is a set of irreducible elements in $D$, then the set of elements in $D$ invertible in $D[S^{-1}]$ is exactly $SD^\times$.
(Such a description does not pass to the case of UFDs: in the polynomial algebra $K[x,y]$, $1/x\notin K[x,y,y/x]$; more generally it fails in an arbitrary non-principal UFD.)
Next, for $D$ an arbitrary domain, one can consider the set of localizations of $D$, that is, the subset $\mathcal{L}'_D$ of $\mathcal{D}$ consisting of those $D[S^{-1}]$ with $S\subset D^\times$.
Such an $S$ can always be chosen to be "saturated" in the sense that it is a submonoid of $D^\times$ satisfying $ab\in S$ implies $a,b\in S$. (About "unitary" I find it hard to find a worse choice of terminology, since "unitary" is widely used in the sense "contains 1".)
The above description of saturated submonoids of $D^\times$, namely as indexed by subsets of $I_D/D^\times$, holds for arbitrary UFDs. It's actually purely a property of the monoid $(D,\times)$, as is being a UFD ($D$ is a UFD iff $(D,\times)$ has an absorbing element $0$, the complement $D\smallsetminus\{0\}$ is a submonoid, and $D\smallsetminus\{0\}$ is direct product of a group with a free monoid.)