Fingerroot is a type of ginger; but if you are not a fan of Eastern cooking, that's a bit misleading. Boesenbergia rotunda, a.k.a. Chinese keys, lesser galangal, Chinese ginger, and a bunch of other names, is a particularly mild form of ginger, in which the 'ginger' flavor is not especially pronounced. The flavor is variously described as earthy, peppery, a bit sweet, a bit herbaceous, slightly bitter, and 'medicinal'. These descriptions should be taken with a grain of salt, as taste is subjective, and many are culturally influenced; for example, fingerroot is a common medicinal plant in many Asian cultures, so the 'medicinal flavor' may be a self-fulfilling descriptor. Personally, I think the earthy flavor is most prominent, but I've only eaten the mature root cooked, which is not the only way to eat it.
It is most common in -- and indigenous to -- China and Southeast Asia, where the rhizome is eaten both raw and cooked. Westerners might be most familiar with it as a main ingredient in namya curry, or pickled and used for flavoring. While it is not common to see it raw and unprocessed in America, it looks, unsurprisingly, like bunches of finger-like roots, much less irregular in form than most ginger species. If one didn't know what it was, it could easily be mistaken for an underdeveloped white radish, or a stunted albino carrot.
As with any cultivated ginger, there are multiple cultivars with variation in flavor and color. The plant itself looks a lot like most ginger plants, with wide leaves and a small white or pink flower; however, be warned, if you are familiar with decorative ginger, this is not that.
Finggerroot is claimed to have a number of medicinal uses, including being used as an male aphrodisiac. It has also been documented to have at least mild anti-allergic, antibacterial, anticancer, anti-inflammatory, antioxidant, and anti-ulcer effects, and is also used to help wound healing. However, it should be noted that many of these benefits are claimed for ginger in general, studies are limited in scope and often limited to rodents, and some of the claimed benefits have scientific backing on the order of "antioxidants can help prevent cancer, and ginger has antioxidants". Please note that people who eat finggerroot on the daily still get cancer, ulcers, and everything else, and check with your doctor before going full herbal on your body.
A full list of fingerroot's various names is not feasible, but in Thai it is called krachai (กระชาย), in Cambodia it is called k'cheay (in Khmer, ខ្ជាយ), in Indonesian it is temu kunci; it will most likely be called ao chun jiang in China. In Japanese it is gajutu, in Lao kneng kieng, and in Vietnamese cu ngai.