Since nobody else answered this question, I'll give it a try from the point of view of mirror symmetry. This is probably a very backwards way of telling the story and there are maybe better answers from, say, the point of view of calibrated geometry. A Bridgeland stability condition consists of a group homomorphism:
$$ Z : K(D) → C $$
and a "slicing" $P$ of $D$ such that if $E \neq 0$ in $P(\phi)$ then Z(E) = $m(E)e^{iπφ}$ for some $ m(E) ∈ R>0 $. See Definition 5.1. of the paper which began this subject: http://annals.math.princeton.edu/wp-content/uploads/annals-v166-n2-p01.pdf
Just as for ordinary stability conditions in geometric invariant theory, there are notions of stability and semi-stability which give us a notion of moduli spaces of objects. There is a conjecture that the derived Fukaya category of the symplectic manifold X carries a Bridgeland stability condition such that Floer theoretically unobstructed special Lagrangians correspond to the semi-stable objects.
We can say that the special Lagrangian condition picks out a nice geometric representative in an isomorphism class in the Fukaya category. Unfortunately, this is probably only a partial truth since there will probably be semi-stable objects which do not correspond to smooth Lagrangians, but singular ones.