Electrical breakdown in a pulsed asymmetric dielectric barrier discharge between a glass-covered mesh electrode and a grounded metal electrode in the air at atmospheric pressure is investigated. Volume discharge forms between the metal tip and the dielectric surface and spreads over the dielectric surface. Breakdown and discharge behaviors depend on the polarity of the charged electrode covered with glass compared to the metal rod electrode. In the case of the dielectric cathode (covered mesh), volume discharge features a stronger and longer-lasting emission. Volume discharge is weaker with outstretched surface discharge developing on the opposite glass electrode sustained by the embedded mesh when the metal rod functions as a cathode. The development and spatial distribution of the surface discharge depend on the relative polarity of the dielectrics caused by the charge deposition of the preceding discharge and is independent of the polarity of the applied high voltage. The discharge emission is brighter for the metal cathode and dielectric anode than for the metal anode, with a branching discharge developing and spreading in a star-like structure along the embedded grid, while a ring-like structure was observed for the metal anode and dielectric cathode. The duty cycle influences the discharge development and properties through the effects of the gas phase and surface pre-ionization.