Absolutely vital core races in D&D

[Aeolius]

For the average campaign, I would stick with the core races as presented in the 1e PH. I rarely run average campaigns, however. In my current undersea campaign, none of the core races are suitable for use as PCs, as they all breathe air. My next campaign concept involves small forest animals used for PCs.

 

[Mustrum_Ridcully]

For D&D, it would be:
Human
Elf and Eladrin - it can be one race with two focuses, or two races, I don't care. The magical and wood elf types just need representation.
Halfing
Dwarf

For any fantasy type, I wouldn't put it down to any specific race, with a caveat - If you don't have humans, have one or more races that basically look like humans, for example:
"North People", "South People", "East People", "West People" or generally "[insert fantasy culture based on real world culture] people"

 

[roguerouge]

I voted for the big four, as that's the big tent needed for the hobby. But for my game, thinking about it, I'd off the dwarves and the elves. Dwarves are invariably played very boring at my tables, when they're played at all. And the long life span of elves makes for huge headaches when it comes to plotting and world design.

So, I'd go with:

Human: ease of identification, everyman status of the races
Halfling: classic fun race
Gnome: the fae race
Changeling: allows the player to investigate issues of identity, "Who am I when I can seem to be anyone?"
Warforged: see above, re: identity issues

 

[Wombat]

Human, (generic) Elf, Dwarf.

The rest are frosting.