5th edition starts with many races, re-vamps the idea of subraces, takes another pass at balancing races while maintaining classic flavor, builds its new mechanics into the races, and nudges the player towards certain norms.
The pool of “standard” PC races has fluctuated over the D&D editions, and 5th edition starts with almost all of them. Dragonborn made it back. Half-orcs are still in. More setting-specific races like warforged and half-giants are not in, but you surely have a dozen homebrew versions online to tide you over until they become official again.
The only big surprise for me was that tieflings made it in but aasimar/devas did not. You get the evil-blooded race but not the good-blooded one. I guess more players were excited about anti-heroes, dark pasts, and stories of redemption than about purer champions of the light. Maybe trying to draw in some White Wolf players? It seems representative of the zeitgeist, A Song of Ice and Fire where white hats get shot off. Batman anchored the New 52, and playing Superman and Captain America straight is refreshingly surprising. I do not really have a soapbox here, more of a rumination that sometimes D&D is all about “no evil PCs!” and other times encourages a bit of a dark streak. Maybe that is productively channeling it, “here is how you get your dark loner who is still a hero who cooperates with the other PCs.”
Oh, and no cat people or vampires. While we are checking off popular boxes: not yet.
5th edition has an elegant approach to subraces that is mechanically similar to what we have seen in other editions but cleaner in its presentations. There is a core to each race, and then most have a “specialization” of sorts. This is a dwarf, and you can be a mountain dwarf or a hill dwarf. You can be a forest gnome or a rock gnome. You can have a whole menu of elves. There are so many elf variations that they are still grouping sub-races within the sub-races (with no mechanical differences), so there are aloof high elves and friendly high elves. The general format is that your base race is +2 to something with some bonuses, while your subrace is +1 to something with some smaller bonus.
The history of human mechanics is interesting. It used to be that non-human races had pluses and minuses, while humans were average. We remember that no one likes minuses, so now all races get pluses, and humans get broader pluses. Non-human races are the best at something and humans are either flexible or just above-average at everything. The base rule in 5th is that humans get +1 to everything (and that is their whole bonus). That seems very strong to me when stats cap at 20 and you might need to save against any stat, not just the three that previous editions had. I have not though deeply about the variant option of fewer bonuses and getting a feat and a skill, which was a very strong bonus in 3rd edition. Given that several feats include +1 to a stat, that could let humans choose their own +2 & +1 that the other races have pre-defined, with a small bonus and a skill. Hmm, that sounds weaker on average but possibly useful for an optimized build.
Humans miss the assorted bonuses that the other races get, but we note those might be just as worthless as +1 to your dump stat, and I wonder about the lack of synergy in 5th. 5th edition builds proficiency and advantage into its races. What if you play a dwarf fighter and now have two sources of axe proficiency, or a gnome wizard with two sources of a saving throw bonus? You just get the one. This seems to encourage playing against type or is perhaps a balancing factor to keep +2 to Int from dictating The One True Wizard Race. Previous editions provided for stacking or typed bonuses, while “has advantage on/proficiency with” does not stack.
Finally, the PHB arranges races into two groups to steer players towards the more standard races. Humans, dwarves, elves, and halflings get presented first. These are the core races, the ones that were main characters in The Lord of the Rings. Half-orcs, gnomes, and tieflings are less common. I doubt that will keep parties from having a hodgepodge of races, but it seems a polite way to indicate a core and fringe. Also, the second tier races have fewer options like subraces.
I sometimes wonder that gnomes continue as a race. Some of that must be for historical reasons, maybe an old school love (or hatred) of illusionists and/or tinker gnomes. The tinker gnome ability strikes me as 2/3 useless, but maybe I am missing some hidden awesomeness there; the illusionist ability seems very strong for the creative player who can think of good uses for nigh-unlimited minor illusions. I look at that and want to play a gnome, maybe a gnome conjurer who has an unlimited supply of fake (but real-looking) illusions and solid (but fake-looking) objects. Fun with pit traps!