While shell-like, Crab-like, and composite supernova remnants (SNRs) represent well-defined classes, there is another group of SNRs with different X-ray and radio morphologies: shell-like in the radio and centrally filled in the X-ray (with little or no limb brightening). We present the results of a systematic X-ray study of 19 such remnants, using ROSAT and ASCA data. We show that they have physical and environmental properties distinct from the other categories, and we propose that they form a unique class that we term ``mixed-morphology'' supernova remnants. We found two prominent distinctions of this new class: their dominant X-ray emission mechanism is thermal despite their X-ray morphological similarity to Crab-like SNRs, and their emission arises primarily from swept-up interstellar material, not ejecta. The temperature across each remnant is nearly uniform, and the density and pressure are constant or increase toward the remnant center, contrary to the standard Sedov model. Most of these remnants are interacting with molecular or H I clouds, as indicated in some cases by strong infrared line emission or OH masers. The population represents at least 8% of all Galactic SNRs and as much as 25% of all X-ray-detected Galactic SNRs. This suggests that supernovae often explode in an environment that is conducive to the formation of a mixed-morphology remnant. The two most likely mechanisms for forming these remnants--evaporation of many small interior clouds and interior ``fossil radiation'' evolved in a denser than normal medium--require a reservoir of material with density higher than the typical intercloud ISM.