I find that some lenses are indicated with the letters "ED", referring to some special type of lens element. What are ED elements, and what advantages do they bring?
Note that this question isn't very clear on ED elements.
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ED means extra-low dispersion, referring to a type of glass that to disperses light less than ordinary glass. Dispersion means breaking up light into its constituent colors due to bending different wavelengths of light to different degrees. Because uncorrected dispersion can cause chromatic aberration, ED glass elements can help reduce purple fringing and other chromatic aberrations, by making it easier to correct for dispersion with the other elements of the lens.
ED elements tend to be limited to higher-end lenses because ED glass can be more expensive to manufacture and work with than ordinary optical glass. Canon goes beyond ED glass in some of its L lenses with fluorite, which has the lowest known level of dispersion, but fluorite elements are fragile and extremely expensive to manufacture and incorporate into a lens (especially if very large, as in the Canon EF 1200mm lens).
ED stands for Extra-low Dispersion
It allows lens manufacturers to exploit the difference in refractive index between different lens elements to help reduce colour fringing in a more effective way, with fewer side effects.
Different types of glass have different refractive indexes: different amounts by which light will bend when they enter the glass.
When light of different colours enters glass, the different colours bend by different amounts. When the resulting image has different colours not lining up properly, this is called chromatic abberation or "colour fringing".
A lens assembly has many different lens elements: some where the light enters or exits a convex surface, and some where the light enters or exits a concave surface. At each step, the different colours will either become more spread out or closer together depending on the orientation of that lens surface.
It is possible to "correct" for colour fringing to some extent by adding more lens elements, that reverse the colour separation effect of previous lens elements. This is rather complex, because you don't want to also reverse the other focusing/magnification effects of the previous lens elements too much.
A low dispersion (such as ED) element can help greatly in this, because you can then use the difference in refractive index between the different lens elements to achieve an effect you couldn't otherwise.