All non-human players are restricted in some aspects and gifted in others.The term non-humans is a broad umbrella in OD&D referring to all bipedal, mammalian, roughly man-shaped and man-sized monsters. These includes elves, goblins, and many others. Later editions would further divide these into demi-humans and humanoids; for races friendly to humans or opposed to same (respectively). We prefer this newer nomenclature for ease of use.
Race As Class
Much criticism has been leveled at Moldvay/Marsh/Cook's B/X or Mentzer's BECMI for restricting races to a certain class. However, OD&D did the same thing with its races. By the book, Dwarves (and presumably Gnomes) and Hobbits could only be fighting-men while Elves could only be a unique combination of Fighter/Magic-User. Even more, all were capped at how high they could progress, though it is further notable no player-characters did not progress to the high levels featured in many modern campaigns.
The Non-Human Player-Characters
No player-character race, nor any monster in service to player-characters, had the ability to see in the dark, later known as infravision. This lack, coupled with level caps, have caused many referees to heavily modify this section of the rules.
Humanocentric Campaigns & Level Caps
Why level caps? EGG stated often they were in place to keep the game humanocentric. He reasoned non-human players would quickly come to dominate play, given their various advantages. We will leave evaluation of this reasoning to you, gentle reader.
Dwarves: gain the advantage of very good saving throws, an architecture-based "sixth sense," and special abilities with a certain type of magic war hammer. The last advantage seemed to better belong in a description of the weapon, not listed as a racial advantage. Like all non-human classes save one, Dwarves begin with a number of languages already known. Dwarves are capped at Fighting-Man level 6 but their improved saving throws made them tough opponents.
The OD&D Volume II monster description for Dwarves also notes this race gains certain advantages to fighting Giant monsters. This entry includes to a callout to the CM description of dwarves, which states they can operate equally well day or night ... infravision, perhaps?
Elves: are Fighting-Men and Magic-Users, switching between the two classes and capped at 4 and 8, respectively. Unfortunately, the text is rather vague as to what, exactly, this entails. According to one quote by EGG, he intended Elf players to basically maintain two separate character sheets. There were many questions in referee's minds over the rules as written: how were hit points and hit dice, leveling, saving throws, etc.; handled? One will find many interpretations among referees who played during that time.
Besides the advantage of being able to use magic armor and still act as a Magic-User the elf spoke a number of languages and also gain the advantages noted in the CHAINMAIL rules when fighting certain fantastic creatures. These CM advantages include split-move & fire, bonus damage with magical weapons (especially versus certain fantastic creatures), and become invisible. The first two are repeated in the OD&D Volume II monster description for Elves.
Even further, the monster description for Ghouls notes Elves are immune to the paralyzing touch of ghouls.By implication in the CM rules, the touch of an Elf removes ghoul paralysis.
Halfings: or Hobbits, depending upon your printing. These are Fighting-Men capped at 4th level, a reasonable approximation of the wee folk as presented in JRRT's The Hobbit. Their advantages include a saving throw bonus and deadly accuracy with missiles as detailed in CHAINMAIL. Halflings do not gain extra starting languages due to race.
In what seems an editorial oversight, there is no Volume II monster listing for Halflings, leaving Chainmail as the sole source of additional information. Unfortunately, we found this entry of little help. It notes Hobbits (this term used in CM) are able to blend into the background and fire a stone as far as an archer shoots. Their deadly accuracy is not well detailed in Alternate Combat System terms however: for every two Hobbits firing count three on the Missile Fire table. Most referees granted a "to hit" bonus to normal missile fire from the wee folk.
Other Character Types: though most printings refer to a Dragon as the example? The first printing used Balrog instead. Another casualty of the "cease and desist" letter from the Tolkien Estate. The description is otherwise the same in all printings.