Bat-Mite is a big Batman fan from the "Mite-Dimension" (later stories claimed he was from the fifth dimension, home of Mr. Mxyzptlk). He wears an "ill-fitting costume" and appears to be a "small childlike man". He has near-god-level reality warping via science too advanced for 3-dimensional people (like us) to understand. He idolizes the Dark Knight and sets up odd events to see his hero in action. He can, however, tell when he gets his hero mad, hence causing him to leave. He is not a super villain, but more of a super nuisance similar to Mister Mxyzptlk.
Bat-Mite appeared regularly for five years in Detective Comics, Batman and World's Finest Comics (which focuses on Batman and Superman) . In World's Finest Comics, he teamed up with Superman's foe, Mister Mxyzptlk, a fifth dimensional "imp" with similar powers, four times. When the Batman comic was revamped in 1964, Bat-Mite had 3 appearances before vanishing. Later, Bat-Mite got a one-page story in The Brave and The Bold # 200.
Bat-Mite was almost completely removed from the DC Comic canon, but he did make some appearances. He got his own one-shot Mitefall (parody of Knightfall). He appeared in World's Funnest, where he and Mister Mxyzptlk destroy the pre-crisis Multiverse. Aside from World's Funnest, he had no direct connection to Mxyzptlk. In the anthology comic Bizarro Comics, in the Mxyzptlk's home dimension, the fifth dimension, there are beings similar Johnny Thunder's "lightning bolt" and Bat-Mite. In a letter column, it says Bat-Mite is from the Fifth Dimension. In Superman / Batman # 25, the Joker had fifth dimensional power (from the Emperor Joker storyline), but Bizarro removed it and it took the form of Bat-Mite. "Might" appears in Batman # 672, but this might have been a delusion. In Batman R.I.P., he appears and tries to counsel the Batman of Zur-En-Arrh, a "delusional" personality of Batman, created by Bruce in case of a mind wipe or being driven mad. In Batman # 680, he reveals to Batman he really is just a product of his imagination, but he claims imagination is the fifth dimension. In Superman / Batman # 52, Bat-Mite is seen making a bet with Mr. Mxyzptlk, similar to the one in World's Funnest.