Chromatic aberration is when light of different wavelengths (i.e., colors) do not focus at the same point. This is considered undesirable in camera lenses. This occurs when a material has different refractive indices at different wavelengths of light.
How is this reduced in higher-end camera lenses? What are the variables in the properties of glass that can be managed to reduce this?
Why doesn’t this occur in our eyeballs?
Different materials have different dispersion. That’s the term for the different index of refraction at different wavelengths. So it’s possible to select different lens materials that have different dispersion, and arrange them so that the chromatic aberration of one lens is partially corrected by another. Such a lens system is called an achromatic lens. A 3-lens system using 3 different materials can do even better, and those are called apochromatic lens. When you see a lens described as “ED”, that stands for extra low-dispersion - referring to a type of glass that is used to put together an apochromatic lens.
It does. But the effect is greatest in the peripheral vision where the retina doesn’t have good resolution anyway. And the brain corrects for some of it.
As scr4 has already noted, you can use the different dispersion of glasses (the way that the variation of index with wavelength) to compensate for the different focal distances. The first practical result was the invention of the cemented achromatic doublet in the 1750s, first used practically in naval teklescopes. The story of the invention is an interesting cautionary tale:
Using a collection of lenses, made of different glasses and with different thicknesses and radii of curvature allows you to color-correct better over a larger region, and to correct the focus from other errors and over the entire range of vision. This is why very good camera lenses are so big and bulky and expensive.
Instead of using different lenses, each of one refractive index, you can also use gradient index lenses, in which the refractive index of the lens isn’t constant, but varies continually. This can let you get away with using fewer optical elements. It’s also one of the ways your eye does this.
finally, you can clear up some errors in processing, rather than fixing the image that falls on your sensing surface. Your eye = brain does this, too. So do a lot of modern camera systems.
Do those let you read the writing on the wall from a greater distance? 
And there are also some high-end lenses now that use a diffractive system (I think they call it a “phase Fresnel lens”). Diffraction gratings are inherently dispersive (in fact, that’s generally the whole point of a diffraction grating), but they typically disperse in the opposite direction from most refractive materials. The benefit of these systems is that a diffraction grating is much thinner than a lens, so you can make the whole assembly much lighter and more compact than a refraction-based system.
High-end cameras don’t have a lens–they have an insane stack of differently shaped ones (and an insane price to match.)
Well, re-invention.
I think it’d be hard to see the chromatic aberration in your own eye. You use different cone cells for different colors, and for a white light off axis, its image on your retina should look like a little spectrum, but the red and green and blue cone cells you associate with one particular location will also be arranged along the same spectrum. There’s no reason you’d have evolved any other way.
No engineer looked at the little spectra that appear on your retina when you look at an arrangement of white lights, and said we need to compensate for that. Evolution would automatically have wired you as if to compensate for it, and would have done so in absolute stupidity.
I have a Canon lens that they call their L glass (for “low dispersion”), but I never looked into what that actually meant. It’s a bit over $1000 and has optics that are dramatically better than my kit lens.
There’s also the concept (not sure if it’s commercially used yet) that some extra materials in glass are what cause differences in chromatic aberration, and so lenses where the level of extra material varies with distance from the center of the lens can correct better with fewer lens components - reduce the complexity of high end camera lenses.