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I just had a look at the paper: Maybe this is also stated more explicitly elsewhere, but I think at least the proof of Lemma 6.3 reveals that an ideal sheaf means a torsion free sheaf for which there exists a nonzero map to $\mathcal{O}_X$, that for families, this condition is just imposed fibrewise (as Sasha suggested) and moreover that this is a closed condition. Anyway, I am leaving my answer as it is, covering the smooth case. (And I still have no idea whether the natural map could fail to be an iso in the singular case.)
@Sasha: You are right. I did point out that I didn't know about the singular case (the OP seemed interested in the smooth case also), but it is worthwhile to note that the very definition of $M_I(X)$ via determinants applies only in the smooth case. Thanks.
@Sasha: This viewpoint makes sense, except that for the embedding of $I$ into $\mathcal{O}_X$ to be unique, you need to assume codimension at least 2. This is what 36min complained about with maximal ideals in $\mathbb{Z}$. So we have to restrict to codimension at least 2 to get a bijection. (What I wrote in my answer, then, is basically that this $M_I(X)$ can be defined without reference to ideals, and that the morphism from the Hilbert scheme is in fact an isomorphism, although that is not entirely obvious and probably independent from what Bridgeland is doing, as you say.)
@36min: And to answer your new question 2: With this definition, you can realize M_I as the fibre over $\mathcal{O}_X$ for the determinant map $M\to \mathrm{Pic}(X)$, where $M$ is the Simpson moduli space for stable/torsion free rank one sheaves.
@36min: Yes, I meant to say that moduli of rank one sheaves with trivial determinant is more or less the standard definition. But, on autopilot, I assumed $X$ to be smooth, I guess Bridgeland does not. If singularitites are allowed, I do not know whether the natural map from the Hilbert scheme is an isomorphism. (And even for $X$ smooth it is not entirely obvious.)
No genericity condition is needed: any line intersects the cubic in an effective degree 3 divisor summing to zero, and conversely. So the result is isomorphic to the projective plane (it is also the linear system |0+0+0|, where 0 means the group identity).
