Part of the difficulty comes from the fact that there are two notions of invertible modules over a ring $A$, to which I'll give provisional names and then quote a theorem showing that they are the same .
Abstractly invertible modules
A finitely generated $A$-module $M$ is abstractly invertible if equivalently:
a) $M$ is finitely generated projective of rank one.
b) There exists some $A$-module $N$ such that $M\otimes_A N \simeq A$ as $A$-modules.
If these conditions hold, then $N$ is necessarily isomorphic to the dual $A$-module $M^*=\operatorname{Hom}_A(M,A)$ and the canonical $A$-linear map $M^*\otimes_A M:\phi \otimes m\mapsto \phi(m)$ is an isomorphism.
The isomorphism classes of these abstractly invertible modules form a group, the Picard group $\operatorname{Pic}(A)$, in which multiplication is given by tensor product: $[M]\cdot[N]=[M\otimes_A N]$, the neutral element is $1=[A]$ and the inverse is determined by $[M]^{-1}=[M^*]$, which explains the terminology invertible module.
Concretely invertible modules
Suppose now that $A$ is a domain with field of fractions $\operatorname{Frac}(A)=K$.
A sub-$A$-module $I\subset K$ is concretely invertible if there exists another sub-$A$-module $J\subset K$ such that $I\cdot J=A$ (here $I\cdot J$ is the $A$-submodule generated by the $i\cdot j$'s with $i\in I$, $j\in J$).
Theorem The module $I$ is concretely invertible if and only if it is abstractly invertible.
If this is the case the submodule $J= A:I=\left\{j\in K\middle|Aj\subset A\right\}$ is the only submodule of $K$ such that $I\cdot J=A$.
Moreover the canonical $A$-linear map $I\otimes J\to A:i\otimes j \mapsto i.j$ is an isomorphism, so that $J$ is a concrete incarnation of the inverse of $I$, namely $[I]^{-1}=[J]$
Bibliography An excellent reference is Chapter 2 of Bourbaki's Commutative Algebra.
Edit: generalization
The most general point of view is that of a locally free sheaf $\mathcal L$ of rank one on a ringed space $X$. This means that $X$ can be covered by open subsets $U_i$ such that $\mathcal L|U_i \simeq\mathcal O_{U_i}$.
Isomorphism classes of such sheaves constitute the Picard group $\operatorname{Pic}(X)$ of the ringed space $X$.
This applies to algebraic varieties, schemes, complex spaces, differential manifolds, ...
If $X=\operatorname{Spec}(A)$ is the affine scheme corresponding to a ring $A$, this group $\operatorname{Pic}(X)$ is isomorphic to the group $\operatorname{Pic}(A)$ defined above under the correspondence associating to the sheaf $\mathcal L$ the $A$-module $M=\Gamma (X, \mathcal L)$.