UPDATE: Amazing X-Men #4 gave the confirmation straight from the horse's (Nightcrawler's) mouth. Looks like my theory in the original answer was pretty spot on.
When BAMFs saved Bobby (Iceman) and he saw Kurt, he asked Kurt "You can start with all these BAMFs. What the hell are they"? Nightcrawler replied:
Isn't it obvious? Don't you see the resemblance? They are my flesh and blood. They are all my little brothers.
Upon snide remark by Bobby about Azazel's clearly having been busy, Kurt clarifies:
Though the BAMFs didn't begin life as my kin. They were born in the depths of hell.
Their mother was a giant maggot. Their father a changeling in the form of a face-eating lamprey. It was love at first sight, I'm sure
The babies were forgotten, as soon as they were born. Left to die on the brimstone plains.
But there they lived. There they survived, in the very shadow of the Pandemonium, as scavengers, as gremlins, as things lower than the lowest beast of the Pit.
And it was there my father found them. On one of his raids.
He found them starving, living off the scabs of the damned and the devil's sewage. And so...
...My father fed them.
He fed them all the blood they could swallow. His blood.
Which changed them, remade them in his image, bound them to his evil will.
... and he explained his involvement with them:
... "And that's where I came in. I found one of Azazel's little red soldiers left behind in heaven. And I made him a deal of my own.
A deal my father could never match. ...
When Firestarter questioned how he turned them from red to blue (since he has no blood to give them, being dead), Nightcrawler said they didn't have time for more explanations since they must rush and find the rest of X-Men. All he said was
It wasn't my blood that I promised them.
TL;DR: They seem to be some sort of Hell creatures/demons originally created by or affiliated with Azazel (who's a Marlvel equivalent of Satan); and who currently (in 2013 comics) serve Nightcrawler - (Azazel's son, a recently deceased X-Men mutant).
*Note: this answer mostly is based on "Wolverine and the X-Men" storyline; and does NOT cover all of the ongoing new sequel series "Amazing X-Men Volume 2", which also features both blue and red Bamfs and most likely may contain further details about them beside those I mentioned below.
The reason there is confusion is that there are at least 3 different types of things referred to as Bamfs in Marvel storylines:
Original Bamf character:
It was a Nightcrawler equivalent in a fairy tale "about x-men" told by Kitty Pryde to Illyana Rasputin.
His appearance was based upon that of Kitty's Bamf Doll (which itself was a 'chibi' version of Nightcrawler). The doll was named "Bamf" since that was the sound Marvel uses to describe Nightcrawler's teleportation.
Appearance: Uncanny X-Men #153
... Except that, it later was found out that Bamfs (who looked exactly like the fairy tale one) were beings who lived in alternate dimension - more specifically, Earth-5311.
Nightcrawler & Lockheed met and fought alongside them.
Those Bamfs were benevolent, liked Nightcrawler and were attracted to Kitty Pride.
Appearance: Nightcrawler issue #2
In Excalibur series (Volume 1, #118/#119), similar-looking beings appeared.
But they were markedly different - malevolent (allied with Nightmare; attacked Lockheed and other team members and said they wanted to kill them); had no affinity for Nightcrawler.
People speculated that they were from different alternate dimension, but there was no clear proof either way in the comics. Aside from looks, one more similarity was romantic interest in Kitty Pryde (their motivation for attacks).
Appearance: Excalibur #118, #119
Then, completely different beings (initially called interdimentional gremlins) showed up in Jean Grey School For Higher Learning on the first day of school. The rest of the answer will deal with them (I will call them "New Model Bamfs" or just "Bamfs" from now on):
First appearance: Wolverine And The X-Men #1
- Beast admitted that he let them though during an experiment.
So far, there was no canon information on whether they are from the same alternate reality as original Bamfs or Excalibur bad Bamfs. But there were several details in further issues that shed some light on their nature.
They are very different in appearance from previous Bamfs in Wolverine&X-Men comic:
They look like a real Nightcrawler, not a doll or children's cartoon character
Wear no cloths
Have strange body markings
NOTE: Only in W&X-Men. In Amazing X-Men they changed the design.
Have Nightcrawler-like prehensile tails
Only say 2 things: "Bamf!" and "Whiskey!"
some Nightcrawler similarity (and later affinity for him)
using "BAMF" sound when teleporting
being from another dimension
BUT... these SAME Bamfs infesting Jean Grey school apparently changed the look an design when new artist was brought in for the sequel, Amazing X-Men. They look now a LOT MORE like the old time Bamfs, no body markings, chubby, more cartoonish looking:
- Professor/extreme zoologist Xanto Starblood in Wolverine And The X-Men #7 indicated they weren't real Bamfs as per @Jake Wallace's answer, but that never got explained further as he was dropped into Krakoa's mouth mid-sentence by them (he wasn't killed as Jake said, but he never discussed them in subsequent issues):
"Bamfs. Ms. Pryde thinks you're Bamfs. I wrote my college thesis on the migration patterns of interdimensional gremlins. I've seen Bamfs. I've studied Bamfs. You, my friends, are definitely not Bamfs. You look more to me like--"
Another interesting fact from that slide is that the reason Xanto was involved with the Bamfs was because they grabbed and teleported him away to save Kitty Pride. On their own volition.
- In Wolverine and the X-Men #17 we get one more hint from unspecified teacher, showing that Bamfs are peculiar:
"The BAMFs are more than just a nuisance now. Their Stealing has become rampant. And the items that are gone missing... it's almost as if they're --"
This appears to be unimportant for now, until we get to Amazing X-Men Volume 2's "The Quest for Nightcrawler" storyline.
Later, in Wolverine And The X-Men #22, we get 2 more details:
Argus Panoptes (Eye Boy, the student with many eyes, and whose mutant power is seeing magic and its traces) finds a tent in Frankenstein's circus with many formaldehided beasties. Among them is a jar with the Bamf, labeled "Hell Beast":
The circus is run by a witch called Calcabrina, which young Frankenstein's robotic cube identified as known affiliation of "Az-"... which most likely was Azazel. (in somewhat related datum, Beast in #23 says that the circus's zombies have Stygian dimension energy signatures - a relatedd reference). Now here's the really curious part:
... in #23, page 21, when Calcabrina is about to be killed, she vanishes... with a BAMF sound!.
And... on page 26 of W&X-M #23, we get full confirmation of that: When Calcabrina prays to her "Master" to help her... in "bamfs"... Azazel (aka Satan, and Nightcrawler's father - remember "Hell Beast" jar label in #22?), surrounded by a retinue of mini-Azazel-looking things, that look like red New Model Bamfs! (this will be important later). They appear to be his creations/familiars/servants, just as demons with similar appearance are Satan's servants in some Christian Hell descriptions.
A second place these Azazel demons can be seen is Amazing X-Men Volume 2 #1:
As we can see, he's acoompanied by the Red Bamfs again (please don't be confused by appearance change - as noted above, they changed the Jean Grey School's Blue Bamfs appearance for "Amazing X-Men", and the red and blue Bamfs match.
In # 29, we get a funny but somewhat irrelevant factoid: Bamfs age, and grow a beard as they do!
The page depicts a future (25 years older than main storyline) Wolverine opening a time capsule from his past; surrounded by Bamfs. Both Bamfs and Wolverine looks much older... and sport beards:
- Our next interesting but irrelevant fact is that they can serve as "teleportation hounds" - in #30 (pages 15 and 18) Wolverine bribes them with whiskey and they "smell" places where Philistine teleported and teleport-pull Wolverine there.
And now, we come to most important facts about them! Issue #35.
- Page 11: We see a white-hooded figure with a Bamf on his shoulder send another Bamf into a spaceship with Broo; to save Broo. The figure (who we know from my earlier spoiler in #29 is Nightcrawler) says "It's time to come back" to Broo, showing that he has keen interest in helping X-Men).
As it ended up, Broo's kind and gentle mutant nature was "resurrected" when he bit that sent Bamf); from Page 20, Broo says, referring to Nightcrawler's message to him on page 11:
I remember fighting for a very long time to be myself again. But never making much ground. And suddenly, someone else was there, giving me the push I needed, telling me it was time to wake up. (Beast: "did this vision of yours happen to coincide with some sort of extraneous event?") - Yes, yes it did. As a matter of fact. Strangely enough, it happened, ... it happened when I bit a Bamf".
... with that last phrace bubble followed by a panel showing a Bamf sitting hidden among the books in Beast's office while they converse, looking at them intently.
- And lastly, the page which reveals where Bamfs came from and who sent them: Page 21. We see thefigure that is strongly hinted to be (currently presumed dead) Nightcrawler - blue face and blue tail shown - talking to Bamfs discussing whether they observed X-Men coming to his aid.
This was, of course, the setup for the follow-up series, Amazing X-Men Volume 2, centering on X-Men saving Nightcrawler.
In Amazing X-Men Volume 2 #1, Beast finds out the solution to the mystery of why and what they were stealing, as mentioned in W&X-M #17... they were building a portal (Page 18).
And on page 20, we get another confirmation of their link to Azazel:
Where they come from, ... the sky is on fire, and there's someone they fear. Someone... red. (Rachel Grey, finally able to read their minds)
and after that:
Someone sent them. After US. Someone needs our help.
Someone sent them. ... Someone we know... It's...
...and she gets interrupted by the machine spewing forth Red Bamfs attacking them to keep them away from the portal.
Wolverine and Northstar get pulled through the gate into Heaven, ending up in a fight with Azazel's pirates; while the red Bamfs pull Storm, Iceman and Firestar into Hell through the gate.