We combine our previous optical spectroscopic and photometric analysis of ~1600 stars located in the Orion Nebula Cluster (ONC) with our own and published near-infrared photometric surveys of the region in order to investigate the evidence for and properties of circumstellar disks. We use the near-infrared continuum excess as our primary disk diagnostic, although we also study sources with Ca II triplet emission and those designated as "proplyds." The measured near-infrared excess is influenced by (1) the presence or absence of a circumstellar disk, (2) the relative importance of disk accretion and inner disk holes, (3) the relative contrast between photospheric and disk emission, and (4) system inclination. After attempting to understand the effects of these influences, we estimate the frequency of circumstellar disks and discuss the evidence for trends in the disk frequency with stellar mass (over the mass range <0.1–50 M⊙), stellar age (over the age range <0.1–2 Myr), and projected cluster radius (over the radial range 0–3 pc). We find that the fraction of stars retaining their inner (<0.1 AU) circumstellar disks to the present time is at least 55% and probably no more than 90%, averaged over the entire range in stellar mass and stellar age represented in the ONC and over the entire area of our survey. We find no trend in the disk fraction with stellar age, at least not over the limited age range of the cluster. We find that more massive stars are less likely to have disks, consistent with a scenario in which the evolutionary timescales are more rapid for disks surrounding more massive stars than for disks surrounding less massive stars. We also find that the disk frequency begins to decrease toward the lowest masses, although objects of all masses (including those that appear to be substellar) can have disks. We find that the disk frequency increases toward the cluster center. We then argue, using several lines of evidence, that a large fraction of the disks associated with stars in the ONC are accretion disks. The observed trends with stellar age, stellar mass, and projected cluster radius in the disk frequency may, in fact, be driven primarily by trends in the disk accretion properties. From the magnitude of the near-infrared excess above that expected from pure irradiation disks, we find an accretion disk fraction among the stars identified as having disks of 61%–88%. In addition, approximately 20% of the stars in our optical spectroscopic sample show broad (several hundred km s-1 FWHM) Ca II emission lines, which are features often associated with accretion disk/wind phenomena; another 50% of the sample have Ca II lines that (at our spectral resolution) are "filled in," indicating an independently derived accretion disk frequency of ~70%. Finally, we discuss the near-infrared and optical emission-line properties of that portion of our sample identified from Hubble Space Telescope imaging as having a dark silhouette or an externally ionized structure. This sample, proposed in the literature to have accretion disks, appears to be no different in terms of its stellar or circumstellar properties from the rest of the ONC population. The only feature distinguishing these objects from their ONC siblings thus may be their current (but short-lived) proximity to the massive stars near the cluster center.
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