A detailed study of the brightness, size, spatial distribution, and filling factor of the different umbral inhomogeneities is presented. Many sunspots show multiple umbrae (separated by strong light bridges, SLBs), which behave like independent units. These are termed umbral cores (UCs). From the phenomenological point of view, UCs show two components: (1) a dark area, formed by a diffuse background (DB), with local intensity minima called dark nuclei (DNs), and (2) bright features including umbral dots (UDs) and faint light bridges (FLBs). Slit-jaw images (bandpass 5425±50 Å) and spectra (line Fe I 5434.5 Å) of seven sunspots with 13 separate UCs were acquired using the Swedish Solar Telescope at the Observatorio del Roque de los Muchachos (La Palma) with a spatial resolution of 0".3. Image restoration and inversion of the spectral line profile were applied, with the following results: The most important photometric parameter of the UCs is the mean intensity of their DB (mean background intensity ‾Iback). It is well correlated to the intensity minimum of the diffuse background and to the mean brightness of UDs. The relation between the size of the UCs and their ‾Iback is not straightforward, but UCs with radii <5" are in general brighter than the larger ones. The differences in the brightness of individual umbrae can be explained mainly by variations of ‾Iback rather than in terms of variations of UD filling factor. The linear relation between the peak intensity of UDs and the intensity of the adjacent DB (local background intensity) was confirmed in a sample of more than 1500 elements, and the "true" peak-to-background ratio was estimated as 3. The spatial distribution of UDs is nonuniform. In DNs (whose size is comparable to that of photospheric granules) UDs appear very rarely; they prefer to form clusters and alignments outside the DNs. The mean nearest neighbor distance decreases, and the density of UDs increases with increasing ‾Iback. The filling factor of UDs is in the range 3%-10%, and the size of UDs is similar in all UCs. The typical "true" size is 180-300 km, and it is uncorrelated to the peak brightness of UDs. FLBs are composed of bright grains and dark spaces between them. The bright grains are darker than photospheric granules. The size of the bright grains is similar to that of UDs, and their relative area in FLBs is very close to the fractional area granulum-intergranulum in the quiet photosphere. On the basis of the obtained spectral profiles, working models of temperature stratification are presented for UDs, their adjacent DB, and DNs. On the basis of observational facts the following is proposed: UDs and FLBs are probably demonstrations of some kind of altered convection, with a cell size of about 0".5, modulated by a smoothly varying magnetic field.