We demonstrate an optical-fiber-based source of polarization-entangled photon pairs with improved quality and efficiency, which has been integrated with off-the-shelf telecom components and is, therefore, well suited for quantum communication applications in the 1550-nm telecom band. Polarization entanglement is produced by simultaneously pumping a loop of standard dispersion-shifted fiber with two orthogonally polarized pump pulses, one propagating in the clockwise and the other in the counterclockwise direction. We characterize this source by investigating two-photon interference between the generated signal-idler photon pairs under various conditions. The experimental parameters are carefully optimized to maximize the generated photon-pair correlation and to minimize contamination of the entangled photon pairs from extraneously scattered background photons that are produced by the pump pulses for two reasons: (i) spontaneous Raman scattering causes uncorrelated photons to be emitted in the signal and idler bands and (ii) broadening of the pump-pulse spectrum due to self-phase modulation causes pump photons to leak into the signal and idler bands. We obtain two-photon interference with visibility $>90%$ without subtracting counts caused by the background photons (only dark counts of the detectors are subtracted), when the mean photon number in the signal (idler) channel is about 0.02/pulse, while no interference is observed in direct detection of either the signal or idler photons.