In the past decade, several rapidly evolving transients have been discovered whose timescales and luminosities are not easily explained by traditional supernovae (SNe) models. The sample size of these objects has remained small due, at least in part, to the challenges of detecting short timescale transients with traditional survey cadences. Here we present the results from a search within the Pan-STARRS1 Medium Deep Survey (PS1-MDS) for rapidly evolving and luminous transients. We identify 10 new transients with a time above half-maximum (t_1/2_) of less than 12 days and -16.5>M>-20 mag. This increases the number of known events in this region of SN phase space by roughly a factor of three. The median redshift of the PS1-MDS sample is z=0.275 and they all exploded in star-forming galaxies. In general, the transients possess faster rise than decline timescale and blue colors at maximum light (g_P1_-r_P1_<~-0.2). Best-fit blackbodies reveal photospheric temperatures/radii that expand/cool with time and explosion spectra taken near maximum light are dominated by a blue continuum, consistent with a hot, optically thick, ejecta. We find it difficult to reconcile the short timescale, high peak luminosity (L>10^43^ erg/s), and lack of UV line blanketing observed in many of these transients with an explosion powered mainly by the radioactive decay of ^56^Ni. Rather, we find that many are consistent with either (1) cooling envelope emission from the explosion of a star with a low-mass extended envelope that ejected very little (<0.03 M_{sun}_) radioactive material, or (2) a shock breakout within a dense, optically thick, wind surrounding the progenitor star. After calculating the detection efficiency for objects with rapid timescales in the PS1-MDS we find a volumetric rate of 4800-8000 events/yr/Gpc^3^ (4%-7% of the core-collapse SN rate at z=0.2).