The formal term for the Academy's trophy is an "Academy Award of Merit," so designated by the governing board of the Academy. In addition to those awards, the Academy has a number of special awards that go by different names. The Academy Honorary Award and Academy Special Awards honor those not in official competition. The Academy Scientific and Technical Awards and the Gordon E. Sawyer Award go to those who design the tools of moviemaking (such as cameras or software). The Jean Hersholt Humanitarian Award goes to members of the film industry who have made exceptional contributions to charitable causes. All of them receive the same statuette, which goes by the more affectionate name of "Oscar." (An additional award, the Irving J. Thalberg Memorial Award, is given out as well, but its trophy is different.)
The name Oscar is sometimes attributed to Bette Davis, who supposedly coined it. Her first husband's name was Harmon Oscar Nelson, and the name was reportedly a reference to him (she said the statue's backside resembled him). There's some dispute over this story, however. Harmon often went by the nickname of "Ham" rather than his middle name, and Davis may have been more likely to use that than "Oscar."
A similarly apocryphal story concerns Hollywood journalist Sid Skolsky, who periodically claimed that he gave the statue its nickname. He said he did so to deflate some of the pomposity surrounding the awards, and to make it feel less pretentious.
By far the most prevalent---and charming---story of the Oscar's origins concerns Academy secretary Margaret Herrick. Upon seeing the award for the first time, she supposedly proclaimed, "It looks just like my Uncle Oscar." If the story is true, she was likely referring to her cousin, Oscar Pierce.
The true origin of the Oscar moniker may never be known: It could be any of the above three stories, or none of them. The term was in use very early, however---Walt Disney used it in 1932, just three years after the awards started---and the Academy has always been happy to promote it. In 1939, they officially dubbed the award the "Oscar," and this name has since eclipsed the more formal "Academy Award" in casual parlance.