We describe in detail an improved standard solar model that has been used to calculate the fluxes of standard solar neutrinos. It includes pre-main-sequence evolution, element diffusion, partial ionization effects, and all the possible nuclear reactions between the main elements. It uses updated values for the initial solar element abundances, the solar age, the solar luminosity, the nuclear reaction rates, and the radiative opacities. Neither nuclear equilibrium nor complete ionization are assumed. The calculated solar neutrino fluxes are compared with published results from the four solar neutrino experiments. The calculated 8B solar neutrino flux is consistent, within the theoretical and experimental uncertainties, with the solar neutrino observations at Homestake and Kamiokande. The observations suggest that the 7Be solar neutrino flux is much smaller than that predicted. However, conclusive evidence for the suppression of the 7Be solar neutrino flux will require experiments like BOREXINO and HELLAZ. If the Be solar neutrino flux is suppressed, it still can be due either to standard physics reasons or neutrino properties beyond the standard electroweak model. Only future neutrino experiments, such as SNO, Super-kamiokande, BOREXINO, and HELLAZ, will be able to show that the solar neutrino problem is a consequence of neutrino properties beyond the standard electroweak model.