Diagram from Slayers TRY Collection #2, page 78
Not much explanation is necessary about this race. The main difference between people of this world and the world of Slayers is the fact that the people of Slayers can use magic. In the Slayers world, anybody can use magic. However, there are differences in magic capacity, and just because a person understands the theory behind Dragu-Slave doesn't mean he/she can cast the spell. However, just by rote memorization, anybody can cast a low-level spell like Lighting. A few individuals possess special abilities such as Telepathy, Beast Mastery, and Prophecy.
Elves and Fairies
People of these race are generally more intelligent, have more special abilities, and live longer than humans. They rarely show themselves to man. The reason is because these people have been used in experiments involving magic. Several races have become extinct because of this, and it is understandable why they choose not to show themselves. Even Lina has only met three Elves and a Nylpha. She has also seen Fairy Souls, said to be the souls of elves. People who lead normal lives are highly unlikely to meet people of this race.
Demi-humans and Monsters
Demi-humans have intelligence, but are different physically. The most well-known demi-humans are beast-men, trolls, and goblins. Beast-men even serve as mercenaries. Monsters are beings that live in darkness, like Mazoku, and are physically completely different. There are many groups of monsters, such as slimes (on the lower end) and vampires (on the higher end). No mistake can be made that monsters cause only harm and do no good towards humans.
Excluding Mazoku, dragons are the strongest race. There are several types of dragons. At the top are the Golden Dragons who are called 'Dragon Lords' and have powerful magic and high intelligence. Then comes the Dimos Dragon's, supposedly the strongest dragons, who live only in the Kataart Mountains. Then come the White Dragons who don't have much strength (though they have enough to be a threat to normal humans), but can speak the human tongue. Further down, there are Plasma Dragons, Deep Serpents, Needle Serpents and Mail Serpents, and numerous other types. In the words of Gourry, "They're all just big lizards."
Dragons who can speak the human tongue are typically rather gentle and won't attack people unless provoked. However, lower level dragons that cannot speak are another story. Some will attack if simply eye contact is made. These dragons do not have the ability to cast spells, and are more similar to beasts in that they use brute strength. There are many types of dragons, but on the most part they are not at odds with humans; neither are they allies. Instead, they take a neutral position.
Animals and Plants
These are basically the same as what exist in this world (though in some areas, there are very dangerous plants). The slight difference is that animals can have "bonds" with certain people. Beast Masters can make animals obey their command by force. On the other hand, people who have "bonds" with certain plants and animals, by being friendly with those species, can ask them to do things. For example, wildlife conservation groups can even have the ability to make plants do favors for them. Naga has "bonds" with jellyfish and inchworms.
The only gods in the Slayers world are Flare Dragon Ceiphied and his counterparts, the Earth Dragon Lord, Water Dragon Lord, Fire Dragon Lord, and Sky Dragon Lord. One of these gods is worshipped in each of the various temples and shrines. However, in Lina's world, the Gods' only presence is the the Water Dragon Lord's knowledge, in the form of the Claire Bible, and the soul of the Flare Dragon, residing within Lina's sister Luna. Basically, the Gods are of no use to the average person.
The beasts of darkness that were born from Ruby Eye Shaburanigdu, Demon King of the world. A totally alien race to not only humans, but all other races. They have far superior magic. The way they're born to the way they think, and even what they eat (Xelloss and Shayla who were in the form of humans ate, but it is highly unlikely that the food was digested) are different.
See also: Mazoku (full article)