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.
Oni in Touhou
In the Touhou Project, oni are generally jolly and 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 abducted. Because normal humans were nowhere near an oni's level of ability, humans resorted to sneaky underhanded tactics in order to beat the oni. Tired of being tricked and killed all the time, the oni swore off humans forever and left for a place beyond human reach, or so the legend says.
In Gensokyo, the oni were once the main residents of the Youkai Mountain, and were above the other races such as tengu and kappa in the hierarchy. They were led by the Four Devas of the Mountain. However, they eventually moved into Former Hell after Hell was moved away from the Underworld by the Yamas. Other youkai moved to the current Hell.
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 time would be too much for anyone to handle.
In The Grimoire of Marisa and Double Spoiler, it's shown that the spell cards of oni are unique or in some cases not even danmaku at all. Oni like Suika Ibuki appear to be poorly skilled at using danmaku and uses brute force to win. According to Patchouli Knowledge and Yuugi Hoshiguma, the oni can be weakened by fried beans, while Hieda no Akyuu claims that running a holly branch through a sardine head would prevent oni from approaching, but she also says that it's unknown if it really works.
Kishin in Touhou
In the 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.
A depiction of oni from Visionary Fairies in Shrine