We investigate the quark deconfinement phase transition in the context of binary neutron star (BNS) mergers. We treat hadronic matter using a Brueckner-Hartree-Fock quantum many-body approach and modern two-body and three-body nuclear interactions derived within chiral effective field theory. Quark matter is modeled using an extended version of the bag model. We combine these approaches to construct a new finite-temperature composition-dependent equation of state (EOS) with a first-order phase transition between hadrons and deconfined quarks. We perform numerical relativity simulations of BNS mergers with this new EOS and compare results obtained with or without the deconfinment phase transition. We find that deconfined quark production in a neutron star merger results from matter crossing the phase boundary over a wide range of temperatures and densities. The softening of the EOS due to the phase transition causes the merger remnants to be more compact and to collapse to a black hole at earlier times. The phase transition is imprinted on the postmerger gravitational wave (GW) signal duration, amplitude, and peak frequency. However, this imprint is only detectable for binaries with sufficiently long-lived remnants. Moreover, the phase transition does not result in significant deviations from quasiuniversal relations for the postmerger GW peak frequency. Consequently, the postmerger GW peak frequency alone is not sufficient to conclusively exclude or confirm the presence of a phase transition in a BNS merger. We also study the impact of the phase transition on dynamical ejecta, remnant accretion disk masses, r -process nucleosynthetic yields and associated electromagnetic counterparts. While there are differences in the electromagnetic counterparts and nucleosynthesis yields between the purely hadronic models and the models with phase transitions, these can be primarily ascribed to the difference in remnant collapse time between the two, so they are degenerate with other effects. An exception is the nonthermal afterglow caused by the interaction of the fastest component of the dynamical ejecta and the interstellar medium, which is systematically boosted in the binaries with phase transition as a consequence of the more violent merger they experience.