These have been a buzzword in skin care for several years now, but they're definitely still being pushed in under-eye creams and serums, especially the expensive ones. On the high end, there's idabenone, which is one of the strongest antioxidants currently available. In the drugstore, there's always good ol' vitamin E, C and some retinol (vitamin A), all of which are more common antioxidants. Since the jury's still out on whether antioxidants applied topically even actually do anything long-term, I'd say save your money.
Hyalauronic acid had a big moment about two years ago after its injectable form, Restylane, was found to fill in naso-labial folds. Off-label use involving under-eye injections has since become popular among a certain set. If injectables aren't your thing, topical acids are all over the market these days, with hyalauronic making an appearance in several drugstore brands. Acids can definitely exfoliate and make skin look thicker and more hydrated, but they won't get rid of broken blood vessels or leftover hemoglobin, which cause the blue/black of circles.
Lots of things are vitamins, of necessity, so I'm usually pretty skeptical of these. Almost every eye cream you pick up will claim to have "vitamins and minerals" the same way breakfast cereals do.
Things ending in -xyl/Peptides:
"Xyl" suffixes usually indicate that it's an amino acid blend, often forming another popular ingredient, the peptide. Peptides can form proteins, hormones, and a host of other more recognizable substances, so when you see these, it could just mean that there's milk or soy protein in your cream. Several proteins like these have traditionally been used to soften or strengthen skin tissue, so it could help some under your eyes. But given the obfuscatory labeling, it's hard to know what, exactly, a given peptide actually does (and if it's something you'll be allergic to).
Source: Flickr User Alan Vernon