Origin

Bamfs are a race from another dimension. To date, only Bamf has been identified as a named individual. However, there are dozens (if not hundreds) or Bamfs living on Bamf Island in an alternate dimension and infesting Jean Grey School for Higher Learning.

Development

Cute & Cuddly (1985)

Originally the race of Bamfs were revealed in the first Nightcrawler mini series. In that story Nightcrawler deduced that they were in an alternate reality where Kitty Pryde's fairy tale from Uncanny X-Men #153 was real life.

There the race of Bamfs mostly resembled clones of Bamf (Nightcrawler's equivalent in Kitty's story) who was a walking, talking, teleporting version of the Bamf Doll. The Bamfs were benevolent and fought to help Nightcrawler & Lockheed the Bamfs. They had an island home, where they had their own friendly society and had a clear gender divide.

Nightmare Alliance (1985)

Although identical in appearance, the Bamfs who appearance in Excalibur were notably more malevolent. It is unclear whether the Bamfs who worked alongside Nightmare were from the same dimension that Nightcrawler & Lockheed stumbled into. They displayed some sort of affection for Kitty which seems to suggest that there's still a connection to her imaginative story, yet showed no signs of recognising Nightcrawler as a father figure.

Jean Grey School For Higher Learning (2011)

In the new series that started as a result of X-Men Schism, Beast mentioned that he had accidentally unleashed an infestation of "Interdimensional Gremlins" into the whole building. The creatures that Beast was referring to have become known a Bamfs, even though their physical appearance is significantly more svelte than previous incarnations. They are considerably skinnier than the cartoony Bamfs that Lockheed & Nightcrawler faced, come with prehensile tails and apparently can teleport in a 'bamf' of purple smoke. This time they don't wear any clothes, and have strange markings on their blue skin (similar to those of Alan Cumming's Nightcrawler from the movie X2: X-Men United).

The Bamfs inside the Jean Grey School of Higher Learning have a limited vocabulary. Similar to Pokémon they originally started by only saying their name "Bamf!", they quickly expanded their vocabulary to a gleeful cry of "Whiskey!" demonstrating that they do have the ability to learn & communicate. As of yet they haven't demonstrated any particular signed of good or evil, but do seem to be a source of constant mischief & support any form of chaos. The Bamfs also seem to enjoy targetting Wolverine's personal belongings, including his whiskey and his hidden collection of martial arts films.

Kitty: The only injuries reported this week were minor ones. Mostly from Bamf attacks. The little blue gremlins apparently found their way into Logan's stash of martial arts instructional videos.
I think I speak for all of us when I say I hope your other video stash is a bit more secure.
- Wolverine And The X-Men #4

Aside from being easily influenced by a desire to wreak havoc, these Bamfs don't seem keen to interact with the X-Men or their students. Instead they're usually spotted in the background, or running around. They have yet to make any difference to the plot of any stories, with a stronger focus on comic relief. It is unclear whether they are related to the previous chubbier Bamfs, or if they are from a different alternate dimension.

There is however a theory that they are not actually Bamfs as pointed out by Professor Xanto Starblood.

Professor Starblood: Bamfs, Ms. Pryde thinks your Bamfs. I wrote my college thesis on migration patterns of interdimensional gremlins. I've seen Bamfs, you, my friends, are definitely not Bamfs. You look more to me like --
- Wolverine And The X-Men #7

The school's syllabus includes 'Bamf Hunting Club with Professor Logan' as an Extracurricular Activity.

Notable Story Arcs

Uncanny Coincidence: Nightcrawler Mini Series

Kitty Pryde told a fairy tale to a six-year old Illyana Rasputin which featured a cartoon version of their friend Nightcrawler called Bamf (although bearing a closer physical resemblance to the Bamf Doll). The character in Kitty's story was jusst that, a fictional character. Yet none of the X-Men ever suspected that a character in one of Kitty's stories would closely resemble real inter-dimensional beings.

Nightcrawler and Lockheed first encountered real Bamfs on their swashbuckling adventure in the Nightcrawler mini series. Nightcrawler deduced that they were in a dimension where Kitty's fairy tale was real life. There, the Bamf treated Nightcrawler as a father figure and introduced the X-Man to Bamf Island. While Nightcrawler battled the villainous Shagreen, the Bamf and Lockheed freed all the captive female Bamfs.

New Year's Evil: Excalibur Ongoing

The Bamfs reappeared in the original Excalibur ongoing series over a decade later. This time Lockheed noticed that Kitty's Bamf Doll was behaving strangely. It turned out to be a Bamf in disguise. As more Bamfs appeared, they began to trap the members of Excalibur using the element of surprise; Kitty & Nightcrawler were trapped inside the Holo-Suite, and Douglock was saved from a crushing boulder by Colossus. After the whole team was overwhelmed by a horde of Bamfs, they were transported to Nightmare's realm and subjected to various mental tortures. They Bamfs were a happy audience as they wanted Kitty's undivided love & attention. Nightmare was defeated by Pete Wisdom's strength of mind & spirit, and there was no sign of the Bamfs when Excalibur awoke back in the real world.

School Infestation: Wolverine And The X-Men

Beast: Did I also happen to mention that two days ago I accidentally left a dimensional gateway open and that the entire building is likely infested with interdimensional gremlins?
- Wolverine And The X-Men #1

The new versions of the Bamfs on the first day of school they immediately caused trouble. They started by chasing the school inspectors out of the building, and cheered on the child-members of the Hellfire Club as they assaulted the school. Upon the Hellfire Club's defeat, the Bamfs ran back into the school with their stolen stash of whiskey.