Mister Mxyzptlk was introduced as a "fifth dimension" imp, unbounded by 3-dimensional laws. So, he was able to do seemingly magical things. He used his powers to cause chaos and pull pranks. He explained to Superman he was a jester in the fifth dimension, hench his pranks. He originally plotted to conquer the Earth, but settled on annoying the Man of Steel. His only weakness is if he says "Kltpzyxm" (his name backwards) he'll go back to his home and stay there for 90 days. He tried to counter this weakness, but failed to do so. He originally appeared as a bald, small man with a purple suit and derby hat and a green bow tie.
In the Silver Age of Comics, it was explained that the reason Mxyzptlk could affect Superman was because he was vulnerable to magic. Mister Mxyzptlk was changed for the Silver Age. Althought the hat stayed, his costume was now orange with purple trims and white hair on the side. In Action Comic # 208, his name is spelt differently: Mxyztplk (not Mxyzptlk). It's unknown if this was a mistake or intentional (as he got revamped in this issue). In a notable adventure, Superman ruined Mxyzptlk's campaign as mayor.
It's revealed that the Golden Age Mxyzptlk is from Earth-Two and the Silver Age is from a parallel reality, Earth-One. In the Superboy stories, the Earth-One Mxyzptlk appears as Master Mxyzptlk. The Legion of Super-Heroes (in Adventure Comics #310) were attacked by Mister Mxyzptlk's descendant (also name Mxyzptlk).
Whatever Happened to the Man of Tomorrow?
A more malevolent Mister Mxyzptlk appears as the main villain in Alan Moore's non-cannon Superman: Whatever Happened to the Man of Tomorrow?. He also revealed his "human" form was not his true form, which is a dark purple, giant, roughly humanoid-shaped ceature.
Mister Mxyzptlk appeared in the modern continuity and was so determined to be a thorn in the Man of Steel's side, he gave Lex Luthor red kryptonite for when he was too busy to annoy Superman. In the Emperor Joker storyline, the Joker got 99% of Mxyzptlk's power and turned the Earth into a nightmarish place, but it turned out the Joker couldn't effect Batman (who defined him), which lead to his downfall. The re-powered Mister Mxyzptlk saved some of the Joker's creation such as the new Bizarro. Various other alternate forms of Mister Mxyzptlk appeared, but it's unclear if they're even related to the character. The real Mister Mxyzptlk (now weakened) saved Superman from Ruin's krytonite-weapon, but seemly died.
One Year Later
Action Comic Annul # 10 said Mxyzptlk hasn't been seen in 190 days and his name is pronounced "Mix-Yez-Pittle-Ick".
Mister Mxyzptlk was captured and imprison in the Source Wall by Superboy-Prime, who had been torturing him. Annataz Arataz (Earth-3 Zatanna) freed the imp. When in the fifth dimension, he (fearfully) told his girlfriend, Gsptlsnz, he cannot go back to the third-diemension and that their dimension must be sealed.
Powers and Abilites
Mister Mxyzptlk, as a fifth dimensional being, has reality warping abilites that are so powerful, they seem like magic to us. The only known limits to Mister Mxyzptlk's power is his willpower and if he says his last name backwards, he'll be sent back to the fifth dimension and be force to stay there for 90 days.
Due to the odd spelling of his last name, it has been pronounced various ways. In the 1967 CBS Filmation Superman animated series, his name was pronounced "mix-yez-PITTLE-ik", the offical pronunciation of the time. However, in the 1980's Super Friends (loosely based on the Justice League), his name was pronounced "Mix-Ill-Plick". "Miks-yez-pit-lik" is how the name is usually pronounced. To complicate things even more, his name has also been pronounced "mix-yez-PIT-lek", "mix-yez-PIT-ul-ick" and "mix-yez-pittle-ik". But in the 1990 Superman: The Animated Series, it's pronounced the same as saying "mix, yes, spit, lick". In the Superman Returns video game, he pronounces his name "Mix-yiz-SPIT-Lik". The pronunciation "Miks-il-piti-lik" (with lighty pronounced i's) was used in Smallville television series. Recently, Action Comics Annual #10 stated his name is pronounced "mix-yez-pittle-ik".