Composite supernova remnants (SNRs) appear center -filled in X-rays and have a shell-like radio morphology. I have used ROSAT and ASCA data to study a sample of ~20 composite SNR, six of these in detail, to understand the origin of centrally enhanced X-ray emission. Seven of the sample remnants (W44, W28, 3C400.2 Kes 27, MSH 11 -61A, 3C391, and CTB 1) were found to be clearly mixed morphology (M-type) composites. In all of these, the X-ray morphology is centrally concentrated and the emission is distributed smoothly in the remnant interior. The central surface brightness is a factor of 2-5 times that at the edge. Despite their X-ray morphological similarity to plerions, the dominant X-ray spectral component is thermal (e.g. ASCA spectra show line emission). Another six remnants (W51C, CTA1, W63, HB21, G327.1-1.1 and CTB 104A) are possible M-type composites. The remnants IC443, Kes 79 and HB 3 are similar to M-type composites; however, other physical processes or environment such as reverse shock or large scale inhomogeneous ISM structures are required to explain their morphologies or spectra. MSH 11-54 is definitely not a M-type composite and W49B and 3C397 might not be. In these remnants, the X-ray emission is enhanced by the ejecta from the progenitor heated by a reverse shock. For M-type composites, the temperature of the X-ray emitting plasma is largely uniform across the remnant, and the pressure and density either do not vary across the remnant, or slightly increase radially inwards, contrary to the classical Sedov solution. I have examined several hypotheses which purport to explain the origin of the centrally concentrated and thermal X-ray emission: "fossil" radiation from the hot interior, a reverse shock, a stellar wind, large scale ISM structures, a reflected shock, and evaporating clouds. The hypothesis invoking evaporation of clouds in the SNR interior (McKee 1981; White & Long 1991) appears to be most consistent with the X-ray data. The temperature, density, and pressure profiles agree with this model. The evaporation of clouds within a multi-phase interstellar medium is the source of the X-ray centrally peaked morphology. The clumps remaining behind a SN shock provide a reservoir of material that evaporates to maintain a high density of hot material within the remnant and thus the observed high X-ray emitting masses are well explained. The clouds must be cold and dense, and numerous to enhance the X-ray emission to produce the centrally-filled morphology. Thermal conduction is expected to be saturated for conditions around evaporating clumps. A simulation of the enhanced X-ray emission due to evaporation around a cloud is presented for the cases of both classical and saturated conduction based on Cowie and McKee (1977). For the classical conduction, the X-ray emission is enhanced by less than a factor of 3 for T < 10^7K. However, when the conduction becomes saturated i.e. the heat flux reaches its limiting value, which is often the case around evaporating clumps, the X-ray emission can be enhanced by a factor of 3-20 and the X-ray emission enhancement appears smooth around evaporating clouds. Therefore, saturated evaporation of clumps that survive a SNR shock can explain the centrally-peaked X-ray morphology of composite SNR.