This very incomplete list tries to provide some of the more accessible references on the material. There are many other excellent references, but those often require a higher level of dedication.
Books
Cox, David A., and Sheldon Katz. Mirror Symmetry and Algebraic Geometry, Mathematical Surveys and Monographs. Providence, RI: American Mathematical Society, 1999. ISBN: 9780821821275.
Hori, K., S. Katz, A. Klemm, R. Panharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow, eds. Mirror Symmetry. Providence, RI: American Mathematical Society, 2003. ISBN: 9780821829554.
Gross, Mark, Daniel Huybrechts, and Dominic Joyce. Calabi-Yau Manifolds and Related Geometries: Lectures at a Summer School in Nordfjordeid, Norway, June 2001. New York, NY: Springer, 2009. ISBN: 9783540440598.
McDuff, Dusa, and Dietmar Salamon. J-holomorphic Curves and Symplectic Topology. Providence, RI: American Mathematical Society, 2004. ISBN: 9780821834855.
Papers
Thomas, R. P. “The Geometry of Mirror Symmetry.” Encyclopedia of Mathematical Physics. Elsevier (2006): 439-448.
Polishchuk, A., and E. Zaslow. “Categorical Mirror Symmetry: The Elliptic Curve.” Adv Theor Math Phys 2 (1998): 443-470; arXiv:math.AG/9801119.
Thomas, R. P. “Derived Categories for The Working Mathematician. Winter School on Mirror Symmetry, (Cambridge MA, 1999).” AMS/IP Stud Adv Math 23, AMS (2001): 363-377; arXiv:math.AG/0001045.
Auroux, D. “Mirror Symmetry and T-duality in The Complement of an Anticanonical Divisor.” J Gökova Geom Topol 1 (2007): 51-91; arXiv:math.SG/0706.3207.
