A race's society is defined as much by its biology as it is defined by its history.
What does gender mean in your society?
Gender roles are defined (for most species at least) in large part by how the species reproduces. The need to reproduce places high value on reproductive functions. Different species exhibit these values in wildly different ways.
Human gender roles are complex, and since this isn't an article about gender politics, I'm going to completely ignore them. Instead, let's look at some nice friendly animals.
Lions have a clearly male-dominated society. The alpha male supplants the previous alpha, the systematically kills all of its young, and claims all of the females in the pack as mates. Beyond reproducing, the male doesn't do a whole lot. Hunting and rearing young is primarily left to the females.
Conversely, ants, bees, and other hive insects have a single queen around which the hive revolves. The queen's primary function is reproduction, so the other members of the hive spend their time feeding and caring for her. Most people don't know this, but nearly all ants are female. The very few males spend their exceptionally short lives guarding and reproducing with the queen.
Does your race raise its young?
This may seem like a weirdly cruel question, but some species like sea turtles, salmon, and sharks lay eggs and never see their children again. Salmon die immediately afterward, so you can't really blame them, but after seeing Finding Nemo you would think sea turtles were better parents. Some species of birds lay their eggs in other species' nests for those birds to raise. Most mammals raise their young, especially herd/pack animals like penguins, horses, and wolves.
Species which don't raise their own children may find it difficult to advance technologically. Without educating their own children, they can't build on their race's previous achievements without some sort of racial memory (oral history, written word, genetic memory, etc.). Some species might raise children communally, claiming no specific parentage of any individual children but communally sharing their culture and knowledge. Humans and human-like species might still raise children in family units similar to those common to real-world Earth.
How does your race congregate?
Humans have succeeded as a species in large part due to our tendency to form groups of one sort or another. Early humans formed nomadic tribes which moved around their territory together, and typically reproduced with other members of their tribe or other nearby tribes. As humans became more populous, humans formed larger societies; cities, states, nations, etc. Humans have tried a huge number of organizational and political structures, and certainly haven't settled on any one definitive option.
Your species might be organized into one singular society of some kind, or it might be a collection of distinct groups. Mass Effect's Krogan formed tribes under alpha male leaders, and the tribes very occasionally cooperated for the betterment of the race (though typically they just murdered each other). Mass Effect's Salarians form family guilds which interact through complex contracts.
The answer "it doesn't" might work, but it can be a difficult choice, especially for races which reproduce sexually.
What does your race believe?
This is a big question. Philosophy, religion, and morality can be a huge part of your race's shared personality. Humans historically have believed that there was some form of higher power, typically a god, to whom they are/were subject or subservient. These gods have taken many forms, have many personalities, and are interpreted in any number of ways. While there are frequent similarities and ideas shared between faiths, the core beliefs can be boiled down to "There is a higher power which decides what is right and wrong, and based on my actions I will receive a reward or punishment when I die." This is obviously a huge simplification of religion, but it serves very well as a simple, shared belief for humans.
If your setting is set in the real-world future, humans may still believe in gods or they may have found some new thing to worship. I recall a cult of some sort in Dead Space which worshiped the ancient race responsible for necromorphs, but I'm too easily startled to actually play the series so my knowledge of the lore is terrible. Perhaps humans have abandoned the concept of a higher power, and wholly embraced science.
Your race's shared beliefs may be more than simply religion. Mass Effect's Krogan believe that constant conflict helps them to evolve as a species, and ending the conflict will cause their species to become weak and die. Major portions of Mass Effect's storyline show that to be fascicle, but many Krogan still cling to the belief. Star Wars's Trandoshans believe in a mystical being who keeps a sort of divine kill count, and those with a high enough kill count go to space lizard heaven.
Your race's beliefs obviously don't need to be true. They could be completely impossible, they could be unverifiable, or they could be wholly correct. They could be a single definitive sentence, or they could be a whole moral code passed down through oral history. Choose whatever fits your race and setting.
What does your race like?
Humans like a great many things. We like food, we like money, we like entertainment. We like to build bigger, better stuff. We like competition. We like winning. Your race might like any number of things. Your race might like nothing. Elves in many fantasy settings like art, natural beauty, and advancing a beautiful craft like architecture, sculpture, or metallurgy. Dwarves similarly tend to like mining, crafting of all kinds, and consumption of liquor in various forms.
What does your race hate?
Humans hate being wrong. We hate being challenged. We hate not getting our way. Things your race hate may be minor inconveniences, but even the most minor hate might be enough to send an entire species to war. Consider what is offensive your race, what is anathema to them in every way. My armor is contempt. My shield is disgust. My sword is hatred. In the Emperor's name, let none survive.
How does your race fight?
This question is fairly crude compared to the other aspects which define a race, but it's important, especially in games which may involve a lot of combat. Humans, especially in Europe, have historically fought by lining up on opposite sides of a battlefield, then marching at each other and killing each other until on side ran away. Even the advent of firearms didn't change this tactic: the cavalry charge was a definitive military tactic until World War I, when the prevalence of automatic weaponry and artillery made it obsolete. The introduction of manned flight and warplanes further changed combat to include dogfighting and arial bombardment. Future leaps in technology will likely have similar effects on military tactics.
Part of the reason that humans can sustain these tactics is that humans are numerous, can learn to fight quickly, and reach fighting age in a reasonably short period of time. If humans took 120 years to reach adulthood like Elves in DnD, they likely wouldn't line up to throw their lives onto enemy spears. Species with exceptionally long life spans and slow reproduction should generally place much greater value on their own lives, and therefore must approach war in much different fashion. Where humans are happy to form rows of peasants with pointy sticks, Elves might be more beholden to tactical strikes, hit-and-run tactics, and dueling.
Dungeons and Dragons-style Goblins, at the opposite end of the spectrum, seem to reproduce in litters, and hit fighting age before you realize that you have new neighbors. In some settings, their race is defined by the divine doctrine of "reproduce, then overwhelm your foes with superior numbers" passed down to them by the god of goblins.
While it's not particularly relevant to this article, I want to make a quick mention of Elves in the Lord of The Rings films directed by Peter Jackson. As much as I enjoyed those films, many of the war scenes seem to ignore the fact that Elves are practically immortal, and should therefore place an immensely high value on their own lives. The scene of the great battle against Sauron at the beginning of Fellowhip portrays Elves forming battle lines and getting charged by waves of Orcs. Unless Elves were ludicrously more populous in those days than they were in the timeframe of the LotR novels, this would have taken a devastating tole on the Elf population. A more logical tactic would be for the Elves to form roving tactical strike units which moved in and out to relieve pressure on human soldiers who were better equipped to match the Orcs' numbers.
Who/what does your race fight?
For real-world humans, it's mostly other humans. In a world with only one sapient species, it's hard to really fight anything else, although there have been historical conflicts with neanderthals, and Australia fought a war against Emus less than a century agho. You could argue that we still fight animals, but that's been a fairly one-sided fight for most of human history.
Your race might war amongst itself, it might be at perpetual war with another species, or it might by at war with literally everything else in the universe like humans in Warhammer 40,000.
It's also important to consider why your race fights whatever it does. Maybe it's for territory, or a philosophical dispute. Maybe it's a war for survival. Maybe it's caught in a perpetual cycle of revenge from which no one has had the ability to escape. Maybe they just really enjoy it.
Example 1 - Shadow of Olympus
I don't know much about the role of women in ancient Greek society, but some very quick Googling provides some basics. Greek society typically followed the patriarchal idea of the silent and obedient wife, but there were examples of women (especially widows) who owned property and managed their own households. I really don't like misogynistic settings for tabletop games, regardless of historical accuracy, so I'm going to deviate here and make women equal members of society. Some city-states might expect married women to forgo any personal ambitions in favor of focusing on their children, but for the most part women have rights equal to men, and female heroes are not a weird exception to the rule.
Humans are organized into city-states. There city-states have a major capitol, and their surrounding lands typically include a few small farm villages and lots of open land. War between city states is common, and each city state typically has a few favored gods whose beliefs they espouse most closely, but all Greeks worship the full Greek pantheon. Greeks enjoy war, seeing it as a noble pursuit, and enjoy tales of heroics. They hate monsters, especially the elemental forces which bleed in from the edges of the Olympian world of our setting.
Minotaurs need a bit more detail since they're a new race. Since they're a cursed race, I want them to be second-class citizens in a lot of ways. They have few permanent homes since they can't own land, so they're very disorganized. Males who are reasonably stable establish a family clan, and keep the group together until someone comes along and murders the alpha. Females have equal social status to males unless they are actively nursing children, in which case they become a respected, matronly figure which males defend with their lives. Minotaur children are raised by their mothers, but have no definitive father figure, partially because fathers are so frequently killed.
Minotaurs believe that Poseidon has cursed them, and the sea will swallow them up if they are foolish enough to board a ship. They also believe that one day a great hero of their race will build a labyrinth which will forever protect their race from the predations of the humans. Minotaurs like stability, and they like to be among their own kind. They hate war, and they especially hate poseidon. Because their race is a cursed abomination, some of them like the taste of human flesh despite a culturable taboo against cannibalism. They are immensely strong and resilient, but lack access to education or advanced technology, so they fight with crude weapons or what they can scrounge from battlefields and fallen foes. They fight humans almost exclusively, but occasionally have territorial disputes with monsters.
Example 2 - Space Grease
Humans live in huge, smog-choked cities capped by enormous smoke stacks which serve no discernible purpose except to produce polution. Other than that, they're basically normal humans. I imagine that children are a very rare sight, and teens appear to spring into existence fully-formed. Robots are essentially the same as humans, but with more rigid personalities (and bodies).
Example 3 - Heroes of Tonesvale
Continuing our "superheroes as a race" idea, we can say a lot about superheroic society. Gender clearly isn't an issue, as there are plenty of female superheroes, but depending on your aesthetic style your superheroines may wear weirdly skimpy outfits. I prefer my superheroins to be more hero than eye-candy, so let's say our female heroes wear reasonably modest and practical clothing, especially in those cold winter months.
Superheroes don't really have children, but they do have sidekicks who they "raise" into new superheroes. These sidekicks are typically paired to one hero until they are old and skilled enough to survive on their own.
Superheroes frequently form teams (Super Friends, Avengers, Justice League, X-Men, etc.) to combine their efforts. In response, villains frequently do the same. Heroes typically believe that they must oppose crime and general badness, while villains believe that they can do whatever they want and get away with it, often at the expense of the population at large. Superhereos like nice things like cute animals, giving to the needy, and vigilante justice. Villains like kicking puppies, twirling their mustaches, and robbing banks (or bakeries in the case of Lex Luthor). Obviously, superheroes and villains fight each other.