Fur Elise was written in 1810 and dedicated to Therese Malfatti, a 14 year old who Beethoven unsuccessfully proposed marriage to.
This tune is familiar to many people today for several reasons. It is used by some telephone companies as the music played when the customer is on hold. It is also one of the first classical pieces that beginning piano students learn to play. The song has been featured in many movies as well, including Death in Venice, Kiss Me Stupid, Fearless, and Rosemary's Baby..
On another note, if your computer ever randomly plays Fur Elise or It's a Small World, do not panic. You probably don't have a virus. This is an indication sent to the PC speaker from the computer's BIOS that the CPU fan is failing or has failed, or that the power supply voltages have drifted out of tolerance. This is a design feature of a detection circuit and system BIOSes developed by Award/Unicore from 1997 on.