This is one of the master's most famous works, and he had a lot of famous works. It's a dead certainty that you've heard it, and if you took piano lessons as a child, there's quite a good chance that you've played this. Beethoven composed it on this day in 1810.
The proper title of the piece is "Bagatelle No. 25 in A Minor"; "Für Elise" is the dedication: for Elise. The interesting question is who was Elise? One theory is that the title was mis transcribed; the piece was lost for 40 years, and the original once rediscovered was lot again so all we have is the transcription. In this theory Elise was Therese Malfatti, whom Beethoven proposed to in 1810. She turned him down. Other women have been proposed as the mysterious Elise, most of whom were sopranos who Beethoven knew, but none had as close a relationship with the maestro as Fraulein Malfatti. In the end, her identity is as lost as the original score.