We present a model for the photoevaporation of circumstellar disks or dense clumps of gas by an external source of ultraviolet radiation. Our model includes the thermal and dynamic effects of 6-13.6 eV far-ultraviolet (FUV) photons and Lyman continuum EUV photons incident upon disks or clumps idealized as spheres of radius r(d) and enclosed mass M*. For sufficiently large values of r(d)/M*, the radiation field evaporates the surface gas and dust. Analytical and numerical approximations to the resulting flows are presented; the model depends on r(d), M*, the flux of FUV and EUV photons, and the column density of neutral gas heated by FUV photons to high temperatures. Application of this model shows that the circumstellar disks (r(d) similar to 10(14)-10(15) cm) in the Orion Nebula ("proplyds") are rapidly destroyed by the external UV radiation field. Close (d less than or similar to 10(17) cm) to theta(1) Ori C, the ionizing EUV photon flux controls the mass-loss rate, and the ionization front (IF) is approximately coincident with the disk surface. Gas evaporated from the cold disk moves subsonically through a relatively thin photodissociation region (PDR) dominated by FUV photons and heated to similar to 1000 K. As the distance from theta(1) Ori C increases, the Lyman continuum flux declines, the PDR thickens, and the IF moves away from the disk surface. At d similar to 3 x 10(17) cm, the thickness of the PDR becomes comparable to the disk radius. Between 3 x 10(17) cm less than or similar to d less than or similar to 10(18) cm, spherical divergence and the resultant pressure gradient in the 10(3) K PDR forms a mildly supersonic (similar to 3-6 km s(-1)) but neutral Parker wind. This wind flows outward until it passes through a shock, beyond which gas moves subsonically through a stationary D-type IF. The IF is moved away from the disk surface to a standoff distance r(IF) greater than or similar to 2.5r(d). In this regime, the mass-loss rate is determined by the incident FUV photon flux and not the ionizing flux. However, at very large distances, d greater than or similar to 10(18) cm, the FUV photon flux drops to values that cannot maintain the disk surface temperature at similar to 10(3) K. As the PDR temperature drops, the pressure of the FUV-powered flow declines with increasing distance from theta(1) Ori C, and again the EUV ionizing photons can penetrate close to the disk surface and dominate the evaporation rate. Radio, H alpha, and [O III] observations of externally illuminated young stellar objects in the Trapezium region are used to determine r(IF) and the projected distances, d(perpendicular to), from theta(1) Ori C. The observed values of r(IF) and d(perpendicular to) are combined with the theory to estimate the disk sizes, mass-loss rates, surface densities, and disk masses for the ensemble of extended sources in the Trapezium cluster. Observations of r(IF), d(perpendicular to), and r(d) in HST 182-413 and a few other sources are used to calibrate parameters of the theory, especially the column of heated PDR gas. The disks have a range in sizes between 14 < log [r(d)/(cm)] < 15.2, mass-loss rates of -7.7 < log [(M)over dot/(M./yr)] < -6.2, surface densities at disk edge 0.7 < log [Sigma(r(d))/(g cm(-2))] < 2.5 which imply disk surface densities at 1 AU from the central, embedded star of 2.8 < log [Sigma/(g cm(-2))] < 3.8 and disk masses of 0.002 < M-d/M. < 0.07. Sigma and M-d scale with the adopted ionization time, t(i), which we take to be 10(5) yr. The inferred Sigma(r(d)) for the ensemble of disks suggest that the initial surface density power law of an individual disk, Sigma proportional to r(-alpha), is bounded by 1 less than or similar to alpha less than or similar to 1.5.
