You really stepped into a hornet's nest, Hindu mythology is very complicated. I'll do the best I can and let others come along later and pick it apart.
So it starts with Lord Brahma, the god of creation. But remember that in hinduism everything is you and you are everything, all creation is linked together etc. Lord Brahma willed into existence the Prajapati (Lords of Creation). These are the great sages of Hindu mythology. The list of sages varies from account to account, but generally these are the names associated with the prajapati:
Angiras, Atri, Bharadvaja, Brigu, Daksha, Gautama, Jamadagni, Kashyapa, Kratu, Marichi, Pulaha, Pulastya, Vashishtha, Vishvamitra
(Note about spelling: Sometimes you'll see "Kasyapa" instead of Kashyapa. They are the same, the difference is in the type of Sanskrit. Kasyapa is older and more formal, but the sound is "sh" and Kashyapa is not incorrect. Vs and Bs are also similarly transposed in some names)
They are also sometimes referred to as the septarishi or "seven sages" but which 7 are the septarishi differs between accounts. The rishis (sages) are the first beings to receive divine knowledge and according to some sources it is not possible for modern men to become rishis anymore. However, it is also an honorific title so it is still used (although there's contention as to whether it should be or not) There are several "rishi" titles such as:
Brahmarishi - A rishi of the brahmin (priest) class
Maharishi - literally "Great sage"
Rajarishi - literally "Royal sage" and typically referring to a king who has become a rishi through good deeds and accumulation of divine knowledge.
Devarishi - There is only 1 devarishi I'm aware of named Narada. He was originally a Gandharva turned rishi.
Paramrishi - I have no idea, I'd love to know though.
Shrutarishi - Sushruta is a Shrutarishi but I don't know what it means.
Kandarishi - Jaimini is a Kandarishi but I don't know what it means.
The above rishis are generally listed in order of power, so the brahmarishi are the most potent.
Now back to the septarishi: The offspring of these sages are the major creatures of Hindu mythology.
Kashyapa (who has 13 wives) is the father of humanity, the Asuras, the Devas, Adityas, Apsaras, Danavas, and Nagas.
Pulaha is the father of the Kimpurushas, Salabhas, and Valikhilyas.
Pulastya is the father of the Kinnaras, Rakshasas, Vanaras, and Yakshas.
Now on to the general creatures of Hindu mythology:
Asuras - They're not really demons in the Christian sense, but they are more accurately described as the opponents of the Gods, or Devas.
Adityas - Gods allied with the Devas, but not necessarily Devas. I'm not too clear on the distinction.
Danavas - In Dominions they are Davanas. These deities are allied with the Asuras, but are eradicated by the Devas at some point.
Devas - The Gods of Hindu mythology, and opponents of the Asuras.
Ganas - There's a spell called Host of Ganas. They are actually a body of attendants, usually to Lord Shiva. They don't really classify as a monster type, just a group of lesser deities I guess.
Gandharvas - Gandharvas are male attendants, the counterparts to Apsaras. Gandharvas and Apsaras sing and dance and entertain the Hindu gods.
Guhyakas - A type of demons that are like Rakshasas but somehow distinct.
Kimpurushas - Men with the heads of lions.
Kinnaras - Alternately harpies (bird men) or centaurs depending on the source.
Mandehas - In Dominions they are Mandahas. Giants very briefly mentioned in the Mahabharata.
Nagas - Serpent men.
Rakshasas - The closest thing that Hindu mythology has to Christian demons. They are shapeshifters and can be beautiful or ugly, thin or fat, etc.
Salabhas - Men with butterfly wings.
Suparnas (or Garudas) - Falcon men and mortal enemies to the nagas.
Valikhilyas - Men who are masters of divination by observing the movements of the sun.
Vanaras - Men with monkey tails.
Yakshas - Sometimes referred to as demons, they are more like dangerous (but not necessarily evil) forest spirits. They have the power of invisibility.
Note about gender: The female versions of all these beings have an 'i' instead of an 'a' at the end of the name. So a female Rakshasa is a Rakshasi.
BUT! Some of the stuff in Dominions is not mentioned here. Rakshasas, being the primary antagonist in the Ramayana have many epithets describing them:
Anusara, Asara, Hanusha - Roughly means "slayer"
Ishtipacha - "Stealer of offerings"
Sandhyabala - "Strong in twilight"
Kshapata, Naktanchara, Ratrichara, Samanishada - Roughly means "night walker"
Nrijagda, Nrichaksha - Roughly means "cannibal"
Palala, Palada, Palankasha, Kravyad - Roughly means "carnivorous"
Asrapa, Asrikpa, Kaunapa, Raktapa, Dandasuka - Roughly means "blood drinker"
Praghasa - "biter"
Malinamukha - "glutton"
Karbura - "black faced"
So in the game, an Asara is a type of Rakshasa, or demon.
For people who don't like mythology, it is speculated that kimpurushas, kinnaras, etc. are all different tribes of men that lived in the remote areas surrounding mount Kailasa and were embellished in Hindu mythology. If you ever have any interest in reading the source material, the 2 most important works to read are the Ramayana and Mahabharata (both available at
www.sacred-texts.com). There are LOTS more tribes mentioned in the Mahabharata.
If you have any more questions I can try to answer them, but I can't guarantee their accuracy as I am no expert on Hindu mythology myself.