Elmer's Pet Rabbit is a 1940 Merrie Melodies cartoon starring Bugs Bunny and Elmer Fudd. The short was released on January 4, 1941. It is the first cartoon in which Bugs is given his name (on a title card). It was directed by the legendary Chuck Jones, written by Rich Hogan, animated by Rudy Larriva, and the music was directed by Carl Stalling. It was produced by Leon Schlesinger and the film editor was SFX wizard Treg Brown.
In this cartoon, Elmer buys Bugs Bunny in a pet shop and Bugs Bunny pesters him. Elmer is voiced by Arthur Q. Bryan and Bugs by Mel Blanc.
The music in the cartoon includes a variation on "While Strolling Through the Park One Day" (Ed Haley), arranged by Carl Stalling, performed by Elmer and Bugs. Elmer, of course, has trouble with many of the words, due to his "rounded L and R" speech impediment.

Trivia

Bugs Bunny wears yellow gloves, he has a deep voice, and he has no buck teeth. Though this cartoon was released after A Wild Hare in which Bugs' recognizable "Bronx/Brooklyn" voice characterization appears for the first time, Mel Blanc used the voice from Elmer's Candid Camera in this short.
This is the first cartoon Bugs quoted the famous Groucho line, "Of course you realize this means war!" (not counting Porky's Hare Hunt in which Happy Rabbit, the early Bugs, but not officialy developed, quoted it).
This was the second cartoon for Bugs and the 23rd cartoon that Chuck Jones directed.