Astaxanthin is a red-pink pigment found in various seafoods, and also in the feathers of flamingos and quails. It is structurally similar to beta-carotene (pro-vitamin A) but has some chemical differences which may make it safer.
It seems to reduce markers of oxidative stress, though it's unclear how potent it is and to what extent this translates to health benefits. It could possibly reduce oxidation of LDL and DNA damage, making it potentially good for cardiovascular health and anti-aging, though much more research is needed. It also may improve photoprotection and overall skin health, but again, much more research is needed.
It's currently unknown if it has notable side effects, though large doses of antioxidants have been known to impair some of the adaptations to exercise such as mitochondrial biogenesis. Being a potent antioxidant, it's possible that even the small doses of astaxanthin commonly taken could have this effect, but we won't know until research properly tests it.