Elliptical galaxy morphology I understand what differentiates elliptical galaxies from spiral, lenticular, and irregular galaxies. What is the difference between a compact elliptical (cE) galaxy and a dwarf elliptical (dE) galaxy? Are there criteria which separate these two subtypes?
The basic criterion is that compact ellipticals live up to their name by being compact -- small in size but with a high density of stars -- while dwarf ellipticals arguably fail to live up to their name because they are more extended and diffuse. (Given that "dwarf" generally means "faint" or "low in stellar mass" when applied to galaxies, you could argue that a better set of terms would be "compact dwarf ellipticals" and "diffuse dwarf ellipticals", respectively.)

The figure below (from Janz et al. 2016) shows size (effective radius "$R_{e}$", which is the radius inside of which half the galaxy's light is found) versus stellar mass for a variety of objects. The relevant part for this discussion is the upper right, where you can find regular elliptical galaxies (green plus signs), dwarf ellipticals (green pentagons), and compact ellipticals (red open circles, plus the black open circle labeled "M32", plus the red plus signs in the same region of the plot). From this you can see the dwarf and compact ellipticals have similar stellar masses (compact ellipticals can sometimes be more massive), but the dwarfs are physically larger. This means that, since they have similar stellar masses and thus similar numbers of stars, they are more spread out and diffuse than compact ellipticals, which are small and dense. Compact ellipticals seem to be an extension of the size-mass correlation of normal (higher-mass) ellipticals, so some astronomers argue they are a continuation of the normal elliptical population, while dwarf ellipticals are a deviation.