Impse is the "act of copulating with an imp". Impse first appeared in a piece of fan fiction posted in the Doomworld Forums and was popularized further by the joke WAD Imp Encounter. The name is a play on goatse, a famous shock site.
The original story
As many have noted, the Encounter story is written in such a manner that the absurd nature of the subject matter takes a back seat, in many ways, to the author's commentary on the degradation of mental capacity that occurs to soldiers exposed to prolonged periods of combat and warfare, such as veterans of Vietnam. At the outset of the tale, the Marine is engaged in desperate battle against overwhelming odds, and must resort to his chainsaw in order to defeat the first wave of enemies. The parallels between the Marine's use of the chainsaw - "in, and out ... in, and out, and the dead cacodemons piled up around him" - and the physical act of love, and indeed between the chainsaw and the phallus could not be more explicit. The tale further affirms the mythic connection between orgasm and death at its literal climax, wherein the Marine's joy at achieving sexual fulfillment is quickly overwhelmed with disgust, self-hate, and a desire to destroy the object that perverted his desire, murdering the Imp and trying to convince himself it was just a momentary lapse in judgement.
What was likely intended as a satirical joke about the overindulgence into seriousness that fanfiction writers often employ actually became the perfect piece of fan fiction for Doom, as this psychological deterioration is emphasized during the game's sparse plot-related text. Even Doom's original slogan, "Where the sanest place is behind a trigger," seems to suggest that the Imp Encounter's theme of severe isolation and traumatic stress problems might be valid. The berserk pack that the marine takes might have also affected his decision.
Impse makes an appearance in many Doom WADs, including Action DOOM, MassMouth 2, It Only Gets Worse and the secrets in Deus Vult. On ZDaemon, it experienced some fame as a secret appearing in a small number of ZDCTFMP2 maps, and became best known through graffiti in the MAGHQ WAD, where the first frame of the animation is captioned "Mike loves imps."
There is also an appearance in Last Man Standing on some of the magazines.