One Race To Rule Them All:
The first and most basic option is to just let players be humans. I think a clear distinction in this is that even if a game only lets everyone PLAY as a human doesn't necessarily mean that those are the only race in the world but some times it does. There are a few pros to everyone playing a single race namely a collective heritage and outlook. Sure this doesn't mean that everyone sees the world in exactly the same light, but there isn't the much bigger changes in perspective that a who other race that might live forever or be born in hell could bring to the table. The biggest con is definitely player options as skills talents and personal back stories differentiate each character meaning no one can be "the one elf" and "the one dwarf" of the party. A good bit of this can be mitigated by having different cultures of a single race as options balancing a bit of a happy medium as this can open up player selection while not being as drastic as having a whole different biology. I used humans as the example in this because more often than not if only one race is allow humans are it, but the concepts hold true no matter what the race is as long as it is the only race for people to play.
The Holy Trinity:
Fantasy gaming has had a racial holy trinity for about as long as it has existed: Humans, Elves, & Dwarves. Sure in my book Halflings/Hobbits are just important, but their inclusion is a bit more scattered and while a staple of D&D and Tolkien fantasy they never reached the saturation into the genre as the others. Even within the setting that do use them more often than not they are considered to blend into the world or man and usually lack a land of their own. Anyways, back onto track. There is a lot to be said about using this group as the core of a fantasy game as they actually represent to three pillars most gaming is actual founded on. The actual balance of what is better in a gamer (Magic User, Stealth Character, or Physical Powerhouse) is a discussion for an entire other article, but in my experience no single group is better working more like a massive game of rock paper scissors. These three races actually fit these three groups of heroes both in their ability and culture much more than you might realize. Sure Elves being magic isn't hard and Dwarves and strength isn't a long shot, but the real trick is Humans. They aren't "stealth" per-say, but more "skilled". In any setting where the core three races exist humans are known for their adaptability and quick learning (Halflings are the ones that literally fill the niche of stealth). As I said I could (and probably will) write an entire article on the triangle of power, but it works and there is a reason that these races have become a cliche of the genre. As I said I'm focusing on fantasy due to the games we are currently heading towards, but this concept still holds true in any setting if you re-skin your races and realize in a futuristic setting what magic=technology.
Here is a bit of a bigger look at races as this applies to races that might or might not be playable in your game world, but all the same needs discussing. In most game worlds there is at least one race if not many that are simple always considered the enemy. Orcs, Goblins, Drow, these are all races that GMs basically use as shorthand for "bad guys" with different seasonings. Yes I know there can be heroes of any of these groups, but in most stories they are the exception rather than the rule and when looking as a race as a whole it's the broad strokes not the individuals that are important. Now playing it safe and simply keeping these "monstrous races" as they are is the standard, but sometimes it's fun to play with standards. This is one of those choices that can really set your world apart from the norm if done well. If you do let players choose to be such races the bigger question is then how the world views that races. Are they seen as the instant villains GMs have used for years or is your world different where maybe Orcs take the hardy strength roll from Dwarves and are just considered as normal. Again I could do an entire article about how many of the core "villains" in fantasy are just darker mirrors of the core three races, but that's something for another time. In settings other than fantasy this concept can ring much stronger (such as iin a modern horror game) or be almost non existent (such as a sci-fi game where unless a race is built up as a villain there probably isn't much preconceived image).
All The Rest:
Most games as they get on in age add more and more races to their world and there's nothing with that. Personally I have a soft spot for Gnomes and Goblins, but that's just me. The key to picking up races as to first decide if everyone's allowed in the pool. This will give your players tons of options, but create its own headaches if you aren't careful. Make sure every single race that a player can select has a place in this world. Try to avoid "the only/last of its kind" as unless you have a really great role player turns the race into nothing more than stat bumps and penalties as there is no culture for them to interact with. Be mindful of how all the races view each other and if there is some kind of divide that more or less prohibits two races from being able to work together then it's time to make the hard choice and go one way or the other because I can almost guarantee that two of your players will pick either race and if not right away then down the road that divide will become an issue.
Sure you can play into these conflicts and problems as story arcs, but for GMs just starting out I wouldn't advise it and for those skilled enough to hand it I'm probably not telling you anything you don't already know.