Bamf is an onomatopoeia used when popular X-Man Nightcrawler teleports. The reason given is that the air is rushing into the space where Nightcrawler used to be. Some fans also use the word as a term for teleportation.
In issue 153 Uncanny X-Men, Kitty Pryde told Illyana Rasputin a fairy tale, where there are a race of beings called Bamfs, who look like super deformed version of Nightclawer. An unnamed Bamf helped the actual heroes of the story.
In Nightcrawler vol. 1, Nightcrawler was accidentally sent into an alternate universe that was almost the same as Kitty's fairy tale even the heroic Bamf appears. He also discovered the female Bamfs are look more like female versions of himself than a female version of the Bamfs. Also, the males are flirtatious and lecherous making them annoying when with female Bamfs.
A recurring joke is the appearances of Nightcrawler dolls called "Bamfs". A group of creatures called Bamfs appeared in 2012 X-Men comics. Beast accidentally sent the Bamfs to our world. These creatures look more like the X2 Nightcrawler and nothing like the previous Bamfs. An alien zoologist claims they aren't really Bamfs, but is cut off before he can explain.