Solar-type stars exhibit their highest levels of magnetic activity during their early convective pre-main-sequence (PMS) phase of evolution. The most powerful PMS flares, superflares and megaflares, have peak X-ray luminosities of log(L_X_)=30.5-34.0erg/s and total energies log(E_X_)=34-38erg. Among >24000 X-ray-selected young (t<~5Myr) members of 40 nearby star-forming regions from our earlier Chandra MYStIX and SFiNCs surveys, we identify and analyze a well-defined sample of 1086 X-ray superflares and megaflares, the largest sample ever studied. Most are considerably more powerful than optical/X-ray superflares detected on main-sequence stars. This study presents energy estimates of these X-ray flares and the properties of their host stars. These events are produced by young stars of all masses over evolutionary stages ranging from protostars to diskless stars, with the occurrence rate positively correlated with stellar mass. Flare properties are indistinguishable for disk- bearing and diskless stars indicating star-disk magnetic fields are not involved. A slope of {alpha}~2 in the flare energy distributions dN/dE_X_{propto}E_X_^-{alpha}^ is consistent with that of optical/X-ray flaring from older stars and the Sun. Megaflares (log(E_X_)>36.2erg) from solar-mass stars have an occurrence rate of 1.7_-0.6_^+1.0^ flares/star/year and contribute at least 10%-20% to the total PMS X-ray energetics. These explosive events may have important astrophysical effects on protoplanetary disk photoevaporation, ionization of disk gas, production of spallogenic radionuclides in disk solids, and hydrodynamic escape of young planetary atmospheres. Our following paper details plasma and magnetic loop modeling of the >50 brightest X-ray megaflares.