EDIT:

The story is too good to miss, so I am adding the text to my answer. I know a lot of people don't like to click on links.

It was a snowy Christmas Eve. As the wind howled outside the castle tower, the Wizard was playing his flute by the fireside, meditating. All the final exams were graded, the apprentices had all gone home, and he had spent an intense week working on Lie n-groupoids and a thought experiment which showed that quantum gravity effects made it impossible to measure distances with an uncertatinly less than Planck length. Now it was time to relax! The fire crackled, the wind whistled down the chimney, the flute's melody swirled like the smoke... it was all very peaceful until he heard someone pounding on his door.

"Whoever it is, go away!" shouted the Wiz.

A muffled voice came from behind the door. "It's me! Eric!"

"All the more reason!" growled the Wiz under his breath, putting down his flute and walking to the door after giving the fire a poke.

"I come seeking knowledge!"

"Oh yeah?" said the Wiz, heaving the oak door open on a crack and sticking his nose out. "What exactly do you want to know?"

"All I'm asking for is reasons!" cried Eric.

"Reasons for what?!"

"You keep saying I am just being ignorant, but I'm holding out until I see some honest to goodness reasons why pseudoforms are necessary."

The Wizard began to fume. "Not that again!" he said. He tried to slam the door shut, but Eric had already slid his foot in the crack.

"Ow! You're not getting rid of me that easily!"

"Hmm," said the Wiz and pondered a while. He couldn't help admiring Eric's persistence. He'd once had an apprentice named Oz, whose education was woefully inadequate in every way, but who managed to make real progress nonetheless, thanks to his almost insane persistence. Unfortunately, Oz had gotten swallowed by another universe as part of a disastrous experiment, and so far all the Wizard's attempts to recover him had failed. Out of pity and fond memories, the Wiz decided to give Eric one more chance.
It was Christmas Eve, after all....

Peering through the crack, he started with a sneaky question: "What sort of thing can we integrate over any smooth n-dimensional manifold?"

Eric cried: "An n-form! This is why differential forms are so great! Everything we actually measure, physically, is the integral of some quantity over some region, or curve, or...."

"Wait," interrupted the Wiz. "Before you launch into your personal philosophy again, let's get the facts straight. It's NOT TRUE that you can integrate an n-form over a smooth n-dimensional manifold. What you can integrate is a PSEUDO n-form."

Eric's mouth snapped shut in mid-rant, and then fell open again in shock. "Hmm... you've got my attention now!" he said.

"Good," growled the Wiz. "So, listen up: You can only integrate an n-form over a smooth n-dimensional manifold if it is equipped with an ORIENTATION. You may be so used to this that you've come to accept the orientation as an inevitable prerequisite for integration. But it's not true! Integration of pseudo n-forms works perfectly fine on any smooth manifold, even an unoriented or unorientable one. It's only if you make the mistake of trying to integrate an N-FORM" - he practically spat the term out in disgust - "that you'll need an orientation. And all the orientation does is let you convert your n-form to a pseudo n-form! Correcting one bad move with another...." He trailed off, grimacing at the folly of the world.

"Hmm... this sounds very interesting," said Eric.

"Good. More to the point, it's the truth," said the Wiz.

"Why didn't any of the 10+ books I have on differential geometry and forms mention this?!"

"What do you expect if you read so few books?!" fired back the Wiz. "You get what you pay for. But surely even those miserable tomes said you could only integrate an n-form on an n-manifold that was oriented. No? And if so, surely it was your job to wonder whether it was possible or not to do integrals on an unoriented manifold! You should have pondered a Moebius strip, and asked yourself: `What's to prevent me from doing integrals on this thing?' You'd chop it up into coordinate charts, do the integral on each piece with the help of a partition of unity, and then add up the results.... And after a little thought, you would have discovered the answer: it all works perfectly fine, if you stick in an extra minus sign to describe how your n-form transforms under a change of coordinates whose Jacobian has negative determinant! In other words, you should use the absolute value of the Jacobian, just like the change of variables formula in multivariable calculus tells you to! But when you do this, you've reinvented pseudo n-forms.

And then, after asking around a bit, you'd have discovered that everyone... everyone who counts, that is... has already realized this!"

Eric pondered a moment. "But... but if it's that simple, why don't my textbooks talk about it?"

Grinning evilly, the Wiz replied: "They certainly drop the necessary clues. As for why they don't emphasize this stuff, well, this is just one of those tricks we Wizards use to distinguish the people who think for themselves from those who fall for any plausible line of claptrap. In fact, every time Halloween falls on a full moon - like this year - we get together and agree on what facts like this we will keep secret, precisely to see who rediscovers it for themselves. This year we...." At this point the absent-minded Wiz caught himself. "Whoops! This is too secret for you! Anyway, you just failed one of these tests."

Mortally offended, but (admirably) more interested in the math than his own dignity, Eric replied "You might shoot me, but you are actually starting to make me think that perhaps it is pseudoforms that are more fundamental than regular forms!"

"Great!" said the Wiz. "Actually, this will relieve me from the need to shoot you, or, more likely, turn you into a toad. Anyway, I hope you see now that people who think they're really integrating n-forms are just slightly less naive than the people who think they're really integrating functions. You can only integrate these things if your n-manifold is equipped with extra structure... and this extra structure is only being used to convert these things to pseudo n-forms!"

"Well," admitted Eric ruefully, "I certainly consider myself to be naive. Honestly, I never knew this. I would really like to understand this better. Got any references?"

"References?!" cried the Wiz. "What do you think I am, a walking journal article? I give you all the clues you need to figure everything out for yourself, and you want references?? Here - HERE's your much-beloved REFERENCES!" he said, flinging a fireball at Eric.

The fireball was large and blue-white, hissing furiously. Luckily for Eric, his reactions were quick and he ducked in time to miss being fried. It hit the wall behind him and exploded in a bang. Simultaneously, the Wiz pushed Eric's foot out from the door and slammed the door shut.

Then he opened a small flap and poked his face out for a moment. "Oh, and by the way: Merry Christmas!" He tossed out a small candy-cane and snapped the flap shut.

Eric was left with his questions. Sucking mournfully on the candy, he stumbled down the castle steps and out into the chilly night.

"Oh, oh!" he wailed. "Can't ANYONE give me some references??"

EDIT:

The story is too good to miss, so I am adding the text to my answer. I know a lot of people don't like to click on links.

It was a snowy Christmas Eve. As the wind howled outside the castle tower, the Wizard was playing his flute by the fireside, meditating. All the final exams were graded, the apprentices had all gone home, and he had spent an intense week working on Lie n-groupoids and a thought experiment which showed that quantum gravity effects made it impossible to measure distances with an uncertatinly less than Planck length. Now it was time to relax! The fire crackled, the wind whistled down the chimney, the flute's melody swirled like the smoke... it was all very peaceful until he heard someone pounding on his door.

"Whoever it is, go away!" shouted the Wiz.

A muffled voice came from behind the door. "It's me! Eric!"

"All the more reason!" growled the Wiz under his breath, putting down his flute and walking to the door after giving the fire a poke.

"I come seeking knowledge!"

"Oh yeah?" said the Wiz, heaving the oak door open on a crack and sticking his nose out. "What exactly do you want to know?"

"All I'm asking for is reasons!" cried Eric.

"Reasons for what?!"

"You keep saying I am just being ignorant, but I'm holding out until I see some honest to goodness reasons why pseudoforms are necessary."

The Wizard began to fume. "Not that again!" he said. He tried to slam the door shut, but Eric had already slid his foot in the crack.

"Ow! You're not getting rid of me that easily!"

"Hmm," said the Wiz and pondered a while. He couldn't help admiring Eric's persistence. He'd once had an apprentice named Oz, whose education was woefully inadequate in every way, but who managed to make real progress nonetheless, thanks to his almost insane persistence. Unfortunately, Oz had gotten swallowed by another universe as part of a disastrous experiment, and so far all the Wizard's attempts to recover him had failed. Out of pity and fond memories, the Wiz decided to give Eric one more chance.
It was Christmas Eve, after all....

Peering through the crack, he started with a sneaky question: "What sort of thing can we integrate over any smooth n-dimensional manifold?"

Eric cried: "An n-form! This is why differential forms are so great! Everything we actually measure, physically, is the integral of some quantity over some region, or curve, or...."

"Wait," interrupted the Wiz. "Before you launch into your personal philosophy again, let's get the facts straight. It's NOT TRUE that you can integrate an n-form over a smooth n-dimensional manifold. What you can integrate is a PSEUDO n-form."

Eric's mouth snapped shut in mid-rant, and then fell open again in shock. "Hmm... you've got my attention now!" he said.

"Good," growled the Wiz. "So, listen up: You can only integrate an n-form over a smooth n-dimensional manifold if it is equipped with an ORIENTATION. You may be so used to this that you've come to accept the orientation as an inevitable prerequisite for integration. But it's not true! Integration of pseudo n-forms works perfectly fine on any smooth manifold, even an unoriented or unorientable one. It's only if you make the mistake of trying to integrate an N-FORM" - he practically spat the term out in disgust - "that you'll need an orientation. And all the orientation does is let you convert your n-form to a pseudo n-form! Correcting one bad move with another...." He trailed off, grimacing at the folly of the world.

"Hmm... this sounds very interesting," said Eric.

"Good. More to the point, it's the truth," said the Wiz.

"Why didn't any of the 10+ books I have on differential geometry and forms mention this?!"

"What do you expect if you read so few books?!" fired back the Wiz. "You get what you pay for. But surely even those miserable tomes said you could only integrate an n-form on an n-manifold that was oriented. No? And if so, surely it was your job to wonder whether it was possible or not to do integrals on an unoriented manifold! You should have pondered a Moebius strip, and asked yourself: `What's to prevent me from doing integrals on this thing?' You'd chop it up into coordinate charts, do the integral on each piece with the help of a partition of unity, and then add up the results.... And after a little thought, you would have discovered the answer: it all works perfectly fine, if you stick in an extra minus sign to describe how your n-form transforms under a change of coordinates whose Jacobian has negative determinant! In other words, you should use the absolute value of the Jacobian, just like the change of variables formula in multivariable calculus tells you to! But when you do this, you've reinvented pseudo n-forms.

And then, after asking around a bit, you'd have discovered that everyone... everyone who counts, that is... has already realized this!"

Eric pondered a moment. "But... but if it's that simple, why don't my textbooks talk about it?"

Grinning evilly, the Wiz replied: "They certainly drop the necessary clues. As for why they don't emphasize this stuff, well, this is just one of those tricks we Wizards use to distinguish the people who think for themselves from those who fall for any plausible line of claptrap. In fact, every time Halloween falls on a full moon - like this year - we get together and agree on what facts like this we will keep secret, precisely to see who rediscovers it for themselves. This year we...." At this point the absent-minded Wiz caught himself. "Whoops! This is too secret for you! Anyway, you just failed one of these tests."

Mortally offended, but (admirably) more interested in the math than his own dignity, Eric replied "You might shoot me, but you are actually starting to make me think that perhaps it is pseudoforms that are more fundamental than regular forms!"

"Great!" said the Wiz. "Actually, this will relieve me from the need to shoot you, or, more likely, turn you into a toad. Anyway, I hope you see now that people who think they're really integrating n-forms are just slightly less naive than the people who think they're really integrating functions. You can only integrate these things if your n-manifold is equipped with extra structure... and this extra structure is only being used to convert these things to pseudo n-forms!"

"Well," admitted Eric ruefully, "I certainly consider myself to be naive. Honestly, I never knew this. I would really like to understand this better. Got any references?"

"References?!" cried the Wiz. "What do you think I am, a walking journal article? I give you all the clues you need to figure everything out for yourself, and you want references?? Here - HERE's your much-beloved REFERENCES!" he said, flinging a fireball at Eric.

The fireball was large and blue-white, hissing furiously. Luckily for Eric, his reactions were quick and he ducked in time to miss being fried. It hit the wall behind him and exploded in a bang. Simultaneously, the Wiz pushed Eric's foot out from the door and slammed the door shut.

Then he opened a small flap and poked his face out for a moment. "Oh, and by the way: Merry Christmas!" He tossed out a small candy-cane and snapped the flap shut.

Eric was left with his questions. Sucking mournfully on the candy, he stumbled down the castle steps and out into the chilly night.

"Oh, oh!" he wailed. "Can't ANYONE give me some references??"