I'll fill in a few details here; more can be found in the references that Daniel gave.
Suppose first that $f:X\rightarrow B$ is a special Lagrangian $T^n$ fibration with only
smooth fibres. If we just want to describe the dual as a complex manifold, we do the following. Hitchin showed that $f$ induces two affine structures on $B$. The important one
for us is the one coming from the fact that $f$ is a Lagrangian fibration. Locally choose
submanifolds $\gamma_1,\ldots,\gamma_n$ of $X$ smooth over $B$ whose restriction to
each fibre provides an integral basis for $H_1$ of that fibre. We can use these to
define one-forms $\lambda_i$ on $B$ via $f_*(\omega|_{\gamma_i})$ (where $\omega$ is
the Kaehler form). Here $f_*$ denotes fibrewise integration. These forms are closed
and hence locally there exists functions $y_i$ such that $dy_i=\lambda_i$. One checks
that $y_1,\ldots,y_n$ form a coordinate system locally on $B$. Once the cycles $\gamma_i$ are chosen the $y_i$ are only well-defined up to constants, and we can always change basis, and
as a result the coordinates are well-defined up to affine linear changes of coordinates,
where the linear part of the affine linear transformation must be integral.
Now here is the dual: consider the local system $\Lambda$ of lattices contained in the tangent bundle $T_B$ of $B$, given locally by integral linear combinations
of the tangent vectors $\partial/\partial_{y_1},\ldots,\partial/\partial_{y_n}$. This
gives a lattice in each fibre of $T_B$. Define the dual torus bundle to be $T_B/\Lambda$,
i.e., divide each tangent space out by the lattice generated by the above tangent vectors.
The projection to $B$ gives the dual torus bundle.
Note that we did not need to have a section of the orginal fibration, although the dual
does have a section. It is possible to twist the dual so that it doesn't have a section;
this is discussed in my paper http://arxiv.org/pdf/math/9809072.pdf
Now this dual, which I'll write as $X(B)$, does carry the ``semi-flat'' complex structure.
This is described via complex coordinates $q_i=\exp(2\pi \sqrt{-1}(x_i+\sqrt{-1}y_i))$,
where $x_i$ is the function on the tangent bundle given by $dy_i$.
Now we allow singular fibres, i.e., consider $f:X\rightarrow B$ whose fibres are
only smooth tori over an open subset $B_0\subseteq B$. So we get the dual $X(B_0)\rightarrow
B_0$ and the question is how we compactify this. In particular, if we want to obtain a complex manifold, we need to deform the above semi-flat complex structure via instanton
corrections. We might assume that $f$ is only a Lagrangian fibration, as that is the
only part of the structure we used to get the dual as a complex manifold, and it is easier to construct Lagrangian fibrations. In this case there has been a certain amount of success describing instanton corrections explicitly in terms of counting holomorphic disks on
$X$. This explicit program was initiated by Denis Auroux in the reference Daniel gave,
but so far can only be done for very simple singular fibres and very few singular
fibres.
The point of view taken by myself and Siebert, as well as by Kontsevich and Soibelman
in the paper Daniel referenced, is that one discards the explicit description of the dual
as a complex manifold, as this introduces too many analytic difficulties in general. Instead, Siebert and I replace this with a degenerating family of schemes over the spectrum of a formal power series ring. Most convergence issues disappear, and when the singularities
are of a relatively simple form, a purely algorithmic approach can be taken to constructing this family. A posteriori one expects that the data produced by this algorithmic approach are in fact the instant corrections, i.e., are produced by counts of holomorphic disks
on $X$. This most precise realization of this expectation is in my paper with Hacking
and Keel, http://arxiv.org/abs/1211.6367 There we explicitly construct the mirror of a surface using the Gromov-Witten invariants of the surface, with no restriction on what the expected singular fibres of a Lagrangian fibration on the surface might be.
