We investigated the electron-pairing mechanism in an iron-based superconductor, iron selenide (FeSe), using scanning tunneling microscopy and spectroscopy. Tunneling conductance spectra of stoichiometric FeSe crystalline films in their superconducting state revealed evidence for a gap function with nodal lines. Electron pairing with twofold symmetry was demonstrated by direct imaging of quasiparticle excitations in the vicinity of magnetic vortex cores, Fe adatoms, and Se vacancies. The twofold pairing symmetry was further supported by the observation of striped electronic nanostructures in the slightly Se-doped samples. The anisotropy can be explained in terms of the orbital-dependent reconstruction of electronic structure in FeSe.