This dissertation discusses the algebraic structures of fixed point Floer homologies. Thedissertation is divided into three chapters, and is adapted from two joint papers by the author. Chapter 1 gives a brief review of the fixed point Floer homology. Chapter 2 gives a detailed computation of the product and the coproduct structures on the fixed point Floer homology of iterations of a single Dehn twist on a surface. Chapter 3 gives a direct verification of the closed-string mirror symmetry for nodal curves based on the computations from Chapter 2.