Half-Elves: Half-elves made their debut as a playable race in Supplement I: Greyhawk, but they might as well be a new race as presented here (at least mechanically). The major advantage of being a half-elf is their extreme flexibility when it comes to class: they can be clerics, druids, fighters, rangers, magic-users, thieves, or assassins, as well as having the ability to combine a whole bunch of those via multi-classing. The OD&D half-elf, by contrast, were fighter/magic-users, and that was that. It's interesting to note that multi-class clerics aren't restricted to the use of blunt weapons - is it possible that the clerical weapon restrictions aren't based on belief and religion? No other explanation seems plausible, unless we give half-elves a special exemption for some reason.
Half-elves get a bunch of elf abilities: their infravision and ability to spot concealed and secret doors are just as effective, but their resistance to sleep and charm is only 30%. They get the same list of languages that elves do as well.
Halflings: Once again, players are referred to the Monster Manual for complete details of Halflings: in this case it's almost necessary, because the three sub-races (hairfoot, tallfellow and stout) aren't really described in the PHB at all. To recap: hairfoots are your baseline halflings, tallfellows are a bit bigger and are friendly with elves, and stouts live underground.
Halflings can be fighters, thieves or fighter/thieves. Taking a look at Supplement I: Greyhawk, it seems to me that this is the first time that halflings are permitted to multi-class.
Halflings get the same resistance to magic and poisons as dwarves and gnomes. It's not mentioned here whether their resistance to magic is due to their being "non-magical", but it's possible, as they're not able to take any spellcasting class.
They can speak the languages of dwarves, elves, gnomes, goblins, halflings and orcs. In OD&D they had no base languages listed. The list matches that in the Monster Manual, except that only tallfellows could speak elvish, and only stouts could speak dwarvish.
Stout halflings have infravision, and can also detect sloping passages when underground. There is mention of halflings with mixed blood, and they get infravision to a lesser extent. To be honest, I don't know why you would ever play a hairfoot or a tallfellow: they get no advantages whatsoever.
Halflings get a stealth ability similar to that of elves: their ability to gain surprise is increased when they are alone and not wearing metallic armour.
One thing that halflings have lost from previous editions is their bonus when using missile weapons. I guess you could say that it's been preserved in their +1 Dexterity bonus, but there's no guarantee that a halfling's Dex score will be high enough to affect attack rolls at all.
Half-Orcs: We learn here that orcs are "fecund", meaning that they have a lot of offspring, and are able to crossbreed with a number of different races. Most of these offspring will be indistinguishable from orcs, but about 10% of orc-human offspring will be closer to human stock, and they're the race detailed here as half-orcs.
Half-Orcs can play as clerics, fighters, thieves, assassins, or a multi-class combination. The option of a cleric/assassin is a tantalising one, I must say. Half-orc clerics can also ignore blunt weapon restrictions, which is more evidence that this isn't a strictly religious rule. I can't come up with a good explanation right now, but I'll think about it.
Half-orcs can speak common and orcish, and they have infravision. They don't get a lot of special abilities. The real advantage of playing a half-orc is that they get a +1 bonus to both Strength and Constitution. Alas, they can't get a Strength of 19, but it's still a super combo for Fighters. Their penalty to Charisma doesn't apply to other half-orcs. (Though it does seemingly apply to orcs, which makes sense from a bigoted perspective.)