Oni (鬼) are fabulous creatures from Japanese folklore, similar to Western demons or ogres. They're almost always depicted as beings with incredible strength and power; physically, two long horns are said to grow from their heads. The earliest folktales of oni generally described them as benevolent creatures able to ward off evil and punish the wicked, but over time, oni began to be seen as cruel, malevolent creatures, content only to destroy. "Oni" in Japanese is sometimes used non-literally to describe beings of similar cruelty, such as vampires (吸血鬼 kyūketsuki, literally "blood-sucking oni").
Kishin (鬼神) is a Japanese word combining the characters for "oni" (鬼) and "god" (神). It usually refers to a wrathful deity, a type of divine being in Buddhism or to an exceptionally powerful oni.
Oni in Touhou
In Touhou Project, oni are jolly youkai who love to drink, feast, and compete with humans in any kind of contest, from simple physical fighting to drinking games. Long ago, when an oni found a human that they liked, they tempted that human into challenging them in any way the human desired. Should the human lose, they were to be abducted. Because normal humans were nowhere near an oni's capabilities, they mostly resorted to sneaky underhanded tactics in order to beat the oni. Tired of being tricked and beaten all the time, the oni swore off humans forever and left for a place beyond human reach, or so the legend says.
In Gensokyo, oni were once the main residents of the Youkai Mountain, and were above tengu and kappa in the hierarchy. They were led by the Big Four of the Mountain. However, they eventually moved into Old Hell sometime after the actual Hell was moved away from the Underworld by the yama. In Hell, oni are lead by the kishin to torture the souls of sinners.
Komachi Onozuka mentions that the oni of current Hell are under control and the ones of Old Hell have too many close relationships with the ghosts of sinners, making them not fond of the idea of leaving to the real Hell anytime soon.
With their high level of physical strength and magical power, oni are considered among the strongest beings in Gensokyo. Moreover, they differ from regular youkai and the means and tools used to exterminate them are also different. Since oni have disappeared for a long time, there are no specialists in oni extermination in Gensokyo, and the methods to deal with them have been lost to humans. Because of this, meeting an oni in the current times would spell disaster for most.
In The Grimoire of Marisa and Double Spoiler, it's shown that some of the spell cards of oni are unique, barely being able to be considered danmaku at all. According to Patchouli Knowledge and Yuugi Hoshiguma, oni can be weakened by fried soy beans. Hieda no Akyuu claims that running a holly branch through a sardine head would prevent an oni from approaching, but it's unknown if that would really work.
Kishin in Touhou
In Touhou Project, kishin are powerful youkai who reside in Hell. Each kishin chief commands a number of lesser oni in their torment of the souls who fall into Hell, utilizing a wide array of torture devices. When a hermit or celestial has lived too long, a kishin chief is sent to assassinate them, though most of the living believe that the assassin is a shinigami due to misinformation they have spread. According to Hecatia in Alternative Facts in Eastern Utopia, there are kishin at the top of the organization of Hell that work with yama. It is unknown if these elite kishin are of the same type of kishin that manage Hell's oni or if they are Buddhist wrathful deities
A depiction of oni from Visionary Fairies in Shrine