We have used the Very Large Array (VLA) FIRST survey and the Automated Plate Measuring Facility (APM) catalog of the Palomar Observatory Sky Survey I (POSS-I) plates as the basis for constructing a new radio-selected sample of optically bright quasars. This is the first radio-selected sample that is competitive in size with current optically selected quasar surveys. Using only two basic criteria, radio-optical positional coincidence and optical morphology, quasars and BL Lac objects can be identified with 60% selection efficiency; the efficiency increases to 70% for objects fainter than 17 mag. We show that a more sophisticated selection scheme can predict with better than 85% reliability which candidates will turn out to be quasars. This paper presents the second installment of the FIRST Bright Quasar Survey (FBQS), with a catalog of 636 quasars distributed over 2682 deg^2^. The quasar sample is characterized and all spectra are displayed.