Yaoyorozu no Kami
Yaoyorozu no Kami (八百万の神, Eight Million Gods) are gods belonging to the Shinto religion. Although the term literally means eight million gods, this is not an exact figure—"eight million" here simply means "countlessly many". Gods can be both good and evil and have a wide range of personality and ideals. Shrines are also built for the deity as a place to occupy in order to interact with humans and other creatures.
Yaoyorozu no Kami in Touhou
Most of the gods that appear in the Touhou Project are part of the Yaoyorozu no Kami, though there are a few who belong to different religions.
Shinto Gods in the Touhou Project are "born" from and sustained by faith. Note that "born" can refer to ascension to godhood, as well, such as the case of Mizue no Uranoshimako, who was a human that became a god even after his death, through faith, passing through being a divine spirit. As they are born from human belief, they share some similarities to youkai, and sometimes have been referred as youkai.[ ] However, the two are significantly different, and for the most part can be considered two different "species".  Unlike gods, youkai were generally born from fear. Thus unlike youkai, gods do not do things such as attack humans by default, but instead seek out faith, typically in the form of worship.
Faith from either humans or youkai is valid, and is described a reverence towards existence, a fear of retribution for one's actions, and the will to enrich one's life, spirit, and body, but it doesn't really have to be that serious. Worship and faith can be as simple as just celebrating with your god. 
Shinto gods may split their spirits and manifest where ever called on, thus enabling them to be in many places at once to serve their worshipers. Certain individuals, such as Watatsuki no Yorihime and shrine maidens like Reimu Hakurei and Sanae Kochiya can directly call upon the gods to enter their bodies. Most gods interact with the physical world entirely in this manner. However, there are some gods with a body, which allows them to act independently without being called upon, such as Kanako and Suwako (and likely Hina Kagiyama, Minoriko Aki, and Shizuha Aki). These gods may still split their spirit and be called upon as well.
The Shinto gods are well-acknowledged by denizens of Gensokyo and denizens of the Lunar Capital (of whom some may have actually been the original gods in Shinto Myth, such as Lord Tsukuyomi and his sister). However, in the outside world, faith has instead been placed on various things such as friendship and news.  Thus, like youkai, it is difficult for some gods such as Kanako Yasaka and Suwako Moriya to survive in the outside world as their faith dies out. Presumably more well-known Shinto gods such as Amaterasu still have plenty of faith to go around in the outside world, as there is nothing to indicate their existence is under threat any time soon.
Of note is that loss of faith for a god means a loss of their ability to manifest their power. However, to a god, this is the same as "death". 
Characters under this Bestiary
Yaoyorozu no Kami not original to Touhou
- Perfect Memento in Strict Sense: Eight Million Gods
- Wikipedia entry for Kami
- Silent Sinner in Blue: Chapter 17 — "You can't fight gods and youkai the same way."
- Mountain of Faith: Kanako Yasaka's profile — "Actually, to play together was what gods wished for." "This was exactly the kind of faith Kanako was wishing for."
- Mountain of Faith: Reimu A ending — "There's no difference between faith and the ability to share a nice drink together like this."
- Mountain of Faith: Reimu B Ending — "In the outside world, people's faith in gods was weakened by the appearance of bad cults and religions. However, all gods benefit from all faith in them. Even though faith and entreatment are different things. Humans, in their independence, lost the need for faith. Actually, that's not true. Even now, they have faith deep within their hearts. People believe in the news, brands, companies, and friends. The object of their faith has merely shifted closer. If you're a god who wishes to regain faith from humans... try setting your pride aside and being friendly with them."
- Mountain of Faith: Sanae Kochiya's profile — "When people lose faith in gods, the gods lose their power. They can't manifest their divine virtues. This is the same as death, for a god."