Question: Lets $E\_{\tau}^{\rho}$ be the elliptic curve with complex structure given by $\tau$ in upper half plane and complexified Kahler form $\rho \frac{dz\wedge d\bar{z}}{2}$.( $\rho$ is in upper half plane too)

Then mirror symmetry says that mirror to $E\_{i}^{\rho}$ in A-side is $E^{i}\_{\rho}$ in B-side.(see the paper of Polishchuk and Zaslow)

then what is the mirror for general $E\_{\tau}^{\rho}$ in A-side (i.e. when we change the complex structure on A-side from the one given by $i$ to something else)??

Answer: 
The mirror of $E^\rho\_\tau$ is $E^\tau\_\rho$, as you may have guessed. The reason this is not discussed in, say, Polishchuk/Zaslow is that the derived category does not depend on the symplectic structure, and the Fukaya category does not depend on the complex structure, so for their purposes the parameter $\tau$ is irrelevant.

Btw, another article on "Mirror symmetry and elliptic curves" that you might find interesting is due to Dijkgraaf's, with title as indicated; it is contained in a 1995 volume on "The moduli space of curves" (see e.g. <http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.8.4194&rep=rep1&type=pdf>)
I think the answer is: $E\_{\tau}^{\rho/\tau\_{im}} \leftrightarrow E\_{\rho}^{\tau/ \rho\_{im}}$ where $\tau\_{im}$ is the imaginary part of $\tau$ (a positive number).

Infact the answer of AByer is correct if write everything in the coordinate $(x,y)$ for the point $x+i\tau=X+iY$ on $E\_{\tau}$ and not the $(X,Y)$.