Why is the mirror of resolved conifold the deformed conifold? I often see a sentence like "the mirror of resolved conifold is the deformed conifold" in physics literature. I would like to ask, in what sense is this true? What is known for mirror symmetry for these manifolds in mathematics?
I am aware of the naive duality of K\"ahler parameter (volume of the exceptional $\mathbb{P}^1$) of the former and complex parameter (radius of the vanishing $S^3$) of the latter, but I expect there should be more to say. I would appreciate it if someone could known any good reference.  
Edit:
I have one more question. Why doesn't this situation globalize? A Calabi-Yau 3-fold with conifold singularity may be smoothed either by resolution or by deformation of the singularity, but they are not mirror each other. 
 A: In the mathematics literature, this example is a basic one for SYZ mirror symmetry, going back at least to Gross' seminal paper
http://arxiv.org/pdf/math/0012002v2.pdf
See Chan,Leung,Lau JDG 2012 for a treatment fully within the realm of symplectic geometry. There were also contributions in Seidel and Thomas from the point of view of homological mirror symmetry in http://arxiv.org/abs/math/0001043. 
Edit: Long winded version: One thing to say about your follow up question is that you could say that if we consider the mirror to a nodal variety, there is no reason to expect that it should be the same. So, heuristically, what we expect is that if we degenerate a variety (say to a nodal hypersurface in a toric variety) then the mirror to the nodal variety should have the same number of nodes and that the mirror to the original smooth variety should be given by resolving. Conversely, if we smooth out this mirror, the smoothing will be mirror to some small resolution of the original nodal toric variety. 
Short version: The fact that the mirror of the resolved conifold is the mirror of the deformed conifold is accidental, purely caused by the fact that the node is,in a sense,
self mirror. 
A: This is a supplementary comment to Rhys's post. Batyrev and his collaborators apply the conifold transition to construction of mirror CY 3-folds of CICY 3-folds in Grassmannians in this paper http://arxiv.org/abs/alg-geom/9710022 
Basic idea is that they degenerate the ambient Grassmannian to a Gorenstein toric Fano variety (toric degeneration) and apply the conventional Batyrev-Borisov construction. As the ambient space degenerates, the CY 3-fold inside gets conifold singularities. The mirror 3-fold is then obtained by resolving the singularity of the (Batyrev-Borisov) mirror 3-fold of the singular one.
It seems that Morrison's conjecture works well in this case.   
A: This is not a complete answer, but I think it's relevant, especially given Daniel's comments under the OP.
There is an old conjecture of Morrison that if $X \rightsquigarrow Y$ is a conifold transition, and $X^*, Y^*$ are the mirrors, then there is a conifold transition $Y^* \rightsquigarrow X^*$; the talk to which Daniel linked discusses an attempt to prove this.
However, Morrison's conjecture is not true in general.  In my paper http://arxiv.org/abs/arXiv:1102.1428, I used Batyrev's formalism of reflexive polyhedra to give examples of conifold transitions which are mirror to completely different transitions (which I call 'hyperconifold transitions').  In these examples, the mirrors $X^*$ and $Y^*$ have different fundamental groups, so cannot be connected by any conifold transition.
There is also a more general approach: Mark Gross explained to me how to use the 'local mirror symmetry' picture of http://arxiv.org/abs/math/0012002v2 to show that a particular hyperconifold transition is mirror to a conifold transition.  This generalises easily to all hyperconifold transitions (hopefully I will soon have a preprint out which includes this simple argument), and demonstrates that Morrison's conjecture fails for all these cases.
A: As Daniel Pomerleano recalled, I have done some work on this with Castano-Bernard. The article is posted here 
http://arxiv.org/abs/1301.2930
I can try to summarise the main ideas. Friedman and Tian showed that a set of nodes in X can be smoothed (in the complex category) if the homology classes of exceptional cycles of a resolution satisfy a good relation. Roughly this means that there is a $3$-dimensional chain whose boundary are the exceptional cycles (actually, it's a bit stronger than this...). On the other hand Smith, Thomas and Yau (see reference given by Daniel Pomerleano) have shown that a ``symplectic conifold'' can be resolved (in the symplectic category) if the vanishing cycles (Lagrangian $3$-spheres) satisfy a good relation (i.e. there exists a $4$-chain bounding the spheres). 
In the Strominger-Yau-Zaslow philosophy of mirror symmetry one should be able to write $X$ as a torus bundle (with some singular fibres...)   $E/L \rightarrow B$. Where $B$ is a real $n$-dimensional manifold, $E$ is a $n$-dimensional vector bundle and $L$ is a maximal lattice ($\cong \mathbb{Z}^n$ over a point). The mirror manifold $X'$ is the dual torus bundle $E'/L' \rightarrow B$. 
Now let $n=3$. If one has a one dimensional vector subspace $V_b \subset E_b$, where $b \in B$, then the anhilator of $V_b$ is a $2$-dimensional subvector space $V'_b \subset E'_b$. If you vary $b$ and $V_b$ over some $2$-dimensional object in $B$ then you obtain a $3$-dimensional object in $X$ and on the mirror you have a $4$-dimensional object. This somehow explains the correspondence between "odd cohomology" in $X$ and ``even cohomology'' in $X'$. 
In our paper, we construct some $2$-dimensional objects (which we call tropical $2$-cycles) which in $X$ give a $3$-chain whose boundary is the union of the exceptional cycles and in $X'$ give a $4$-chain whose boundary is the union of the vanishing cycles. This proves that if such tropical $2$-cycles exist the obstructions to smoothing $X$ and to resolving $X'$ vanish simultaneously. 
Our work relies very much on Mark Gross's ``Topological Mirror Symmetry'', which you can find here
http://arxiv.org/abs/math/9909015 
A: As Eric Zaslow pointed out, the quoted statement "the mirror of resolved conifold is the deformed conifold" isn't true (in general). The correct statement should be (at least conjecturally):
"The mirror of a conifold $\bar{X}$ is a conifold $\bar{X}'$; the mirror of resolved conifold of $\bar{X}$ is the deformed conifold of $\bar{X}'$ and vice versa." 
A useful reference is a paper by Smith, Thomas and Yau in JDG.
Caveat: These are not mathematical statements, since there is no mathematical definition of "mirror".
