Recent surveys of the Orion Nebula with the Hubble Space Telescope have revealed a number of stars surrounded by dark silhouettes seen projected against the bright background H II region. In this paper, we present a detailed analysis of HST and ground-based observations of the six most distinct silhouettes. We find a variety of morphologies, all consistent with thin circumstellar disks spanning a range of diameters (50 to 1000 AU) and inclination angles (0 to >80 degrees). The silhouette intensity profiles cannot be fit by standard disk models in which the surface density follows a radial power law with an exponent in the range -0.75 to - 1.5. Rather, the data are best fit by opaque inner disks with exponential edges, and we discuss possible physical origins of this apparent truncation. Masses in the range 6×1026-4×1030 g (i.e., up to 0.002 Msun) are determined for the disks by assuming that the faint light measured from them is background light transmitted through the disk. However, we argue that these are strict lower limits on the true disk masses, as most of this light can be accounted for by PSF blurring and scattering in the HST optical train, and that the present observations are in fact consistent with completely opaque disks. Central stars are seen directly in five of the silhouettes, while the presence of a star is inferred in the sixth, where small reflection nebulae are seen above and below the plane of the near edge-on disk. Optical and near-infrared stellar photometry is consistent with young (˜1 Myr) low-mass (0.3-1.5 Msun) stars, with several showing evidence for excess near-infrared emission from the disk inner edge.