Pinky and The Brain/WMG

Pinky and the Brain are in Purgatory
They have to work off their mortal sins through their constant attempts to take over the world. A bit of an odd way of working off sins, but there you go. Might also explain why the entrance to Hades was just a walk away in "A Pinky and the Brain Halloween".

Pinky is the genius.
The theme song says "One is a genius, the other's insane," but it doesn't say which, and megalomania has "mania" in it. What more need be said? According to Wikipedia, rather a lot.
 * Also note that being insane doesn't necessarily make you stupid, now does it?
 * Pinky's several own attempts at world domination were very nearly successful, stopped at the last minute by none other then The Brain, usually simply because he refuses to look past the fact that it is Pinky doing something, so it must be stupid. Many of Brain's schemes are ruined by his own mistakes, or because he overlooks any advice and reproves action from Pinky which, despite usually being ridiculous, generally turns out to be right. All of this goes to prove that Pinky is a genius, repeatedly foiled by the Brain's insanity.
 * Plus, there was that one episode where Pinky momentarily displayed telekinetic abilities, leaving Brain completely dumbfounded and in awe.
 * The crossover episode "In the garden of Mindy" also seems to hint this, with replacament of Pinky with Mindy leading to the following change in lyrics: "They're Mindy and the Brain, Mindy and the Brain, one is small child, the other's... the Brain", as if "insane" was meant to be atributed to Brain all along.
 * Rebuttal, though I agree with the WMG: If Brain was the insane one, they could have left that bit the same, and the pause could have been to indicate the singer improvising.
 * YES, seconded! Imagining this makes the show even more hilarious.

Pinky deliberately keeps the Brain in check.
In a case of Obfuscating Stupidity taken to extremes, Pinky plays the part of loyal sidekick, but in fact waits for the precise moment to cause Brain's plans to fail -- or, more commonly, to cause Brain to foil his own plan. This rather relies on the "Pinky is the genius" theory, but is not a necessary conclusion thereof.
 * The easiest way is Pinky is the genius (saves the world on a nightly basis) and The Brain is insane (he's a mouse that tries to take over the world dozens of times no matter how often he fails).
 * Really doubtful, in a Christmas Episode Pinky tells Santa Claus that Brain has the world's best interests at heart, and he's not really a bad person. Though you could still make the case that Brain would be a terrible world leader, and Pinky knows this.
 * Well, Pinky is absent-minded, but definately not insane.

Pinky is Adam Savage, and the Brain is Jamie Hyneman.
A myth involving brain-swapping went horribly, horribly wrong. This of course makes the Brain also the Eggman, the Walrus and the Carpenter; unfortunately, mice can't grow moustaches.
 * So he's THE Eggman?
 * Goo Goo G'joob.

Elmyra is the spawn of Satan.
Nuff said.
 * Jossed, I think - we actually see her father in two Tiny Toons episodes. Or is he an incarnation of Satan?
 * What else would you expect from a Duff?

Pinky, Elmyra, and the Brain is All Just a Dream.
Acme Labs was never turned into a Disney Store, Elmyra never got transferred from Acme Looniversity to Chuck Norris Grammar School, and Pinky and Brain aren't living in Elmyra's house in order to hide from Wally Faust - it's all just a bad dream that Brain's having back in the lab, just like the events depicted in "You'll Never Eat Food Pellets in This Town Again" and the Animaniacs short where he was the Hip Hippos' baby.

Pinky, Elmyra, and the Brain is a hallucination.
Or in a worse-case version of the above theory, Wally Faust has already captured the mice, and the events depicted in PEatB are just a hallucination they're experiencing as he experiments on them.

Pinky and the Brain are both geniuses, and both insane.
Brain is definitely the Mad Scientist variety, combining scientific genius with megalomania. Pinky is more of a Mad Artist - he does have his flashes of brilliant insight but also far too many Cloudcuckoolander moments to be all there. (Plus, it's fun to think of Pinky as being dismissive of Brain's scientific endeavors - think of him as a Seen It All Deadpan Snarker and the show becomes that much more funny to watch.)

The citizens of Brain's world are, on the whole, quietly in support of his plans for world domination
Even at his worst, Brain has been shown to be not all that bad and is even willing to admit his mistakes when he sees he's gone too far with something. People realize this and, while they do expect they'll have his eccentricities to deal with, they see him as a decent leader. However, to speak out as such would get one in some trouble, so they settle for letting him go about his tasks largely unopposed and even help him by pretending not to see through his Paper-Thin Disguise.

Pinky is kinky
He loves being bopped around by Brain, and probably finds him a Romantic Abuser. That would account for all the Ho Yay on the show and why he stays with Brain...plus, in the Halloween episode, Pinky is tortured by Mr. Itch (The Devil) and laughs wildly the whole time, even screaming "AGAIN! AGAIN!!" after being shaken in a paint mixer. This later prompts Mr. Itch to inquire of a smoldering, barbequed Pinky, "Isn't there ANY kind of torture you don't like?!" The answer was no, "The only real torture...is being away from my best friend Brain."
 * Fun fact: In the Dutch dub of the series the intro states that they are kinky not dinky (probably because there aren't a lot of words in the Dutch language that rhyme with Pinky.)

Pinky has Hollywood Tourettes.
Only he says nonsense words instead of cusswords. (TROZ!)

Elmyra is the daughter of Elmer Fudd.
The face, the voice, the name--c'mon people! Seems in trying to defy her father's (or "daddy-waddy-head's") hatred of "wabbits" she took it to the opposite extreme.
 * Wrong show, and Jossed anyway; not only is Elmyra's father seen, but the only Tiny Toons character descended from a Looney Toon was Gogo.

Pinky and Brain's personalities are the results of having been experimented on.
"They're laboratory mice/Their genes have been spliced"... And though we don't really get to see this happen, we do get to see the results. Brain became a genius via Flowers for Algernon-esque experiments designed to improve his intelligence, which also gave him a dim view of humanity in general. Similar experiments were tried on Pinky, but they messed with his brain wiring instead of improving him.