We report the discovery of HAT-P-14b, a fairly massive transiting extrasolar planet orbiting the moderately bright star GSC 3086-00152 (V=9.98), with a period of P=4.627669+/-0.000005 days. The transit is close to grazing (impact parameter 0.891^+0.007^_-0.008_) and has a duration of 0.0912+/-0.0017 days, with a reference epoch of mid-transit of T_c_=2454875.28938+/-0.00047 (BJD). The orbit is slightly eccentric (e=0.107+/-0.013), and the orientation is such that occultations are unlikely to occur. The host star is a slightly evolved mid-F dwarf with a mass of 1.386+/-0.045M_{sun}_, a radius of 1.468+/-0.054R_{sun}_, effective temperature 6600+/-90K, and a slightly metal-rich composition corresponding to [Fe/H]=+0.11+/-0.08. The planet has a mass of 2.232+/-0.059M_J_ and a radius of 1.150+/-0.052R_J_, implying a mean density of 1.82+/-0.24g/cm^3^. Its radius is well reproduced by theoretical models for the 1.3Gyr age of the system if the planet has a heavy-element fraction of about 50M_{earth}_ (7% of its total mass). The brightness, near-grazing orientation, and other properties of HAT-P-14 make it a favorable transiting system to look for changes in the orbital elements or transit timing variations induced by a possible second planet, and also to place meaningful constraints on the presence of sub-Earth mass or Earth-mass exomoons, by monitoring it for transit duration variations.