Au-n-octanedithiol-Au molecular junction (Au-SC8S-Au) has been investigated using density functional theory (DFT) combined with the non-equilibrium Green’s function (NEGF) approach. Theoretically calculated results are used to build the relationship between the interface binding structures and single-molecule quantum conductance of n-octanedithiol (SC8S) embodied in a gold nanogap with or without stretching forces. To understand the electron transport mechanism in the single molecular nanojunction, we designed three types of Au-SC8S-Au nanogaps, including flat electrode through an Au atom connecting (Model I), top-pyramidal or flat electrodes with the molecule adsorbing directly (Model II), and top-pyramidal Au electrodes with Au atomic chains (Model III). We first determined the optimized structures of different Au-SC8S-Au nanogaps, and then predicted the distance-dependent stretching force and conductance in each case. Our calculated results show that in the Model I with an Au atom bridging the flat Au (111) gold electrodes and the SC8S molecule, the conductance decreases exponentially before the fracture of Au-Au bond, in a good agreement with the experimental conductance in the literature. For the top-pyramidal electrode Models II and III, the magnitudes of molecular conductance are larger than that in Model I. Our theoretical calculations also show that the Au-Au bond fracture takes place in Models I and III, while the Au-S bond fracture appears in Model II. This is explained due to the total strength of three synergetic Au-Au bonds stronger than an Au-S bond in Model II. This is supported from the broken force about 2 nN for the Au-Au bond and 3 nN for the Au-S bond.