This music is the Nocturne Op. 9 No. 2 in E-flat major by Frederic Chopin. A Nocturne was originally a piece of music to be played in the evening, a similar concept to a Serenade, but it came to mean simply a stand-alone piece usually for piano and with a rich Romantic melody. Chopin was one of the foremost composers of Nocturnes writing 21 in total, and No.2 is his most famous one and has been recorded by many pianists. Below is a video showing the music in a "piano roll" animation, and below that you can play the mp3 file and see the sheet music before downloading. To download the sheet music, mp3 file and midi file use the links in the left-hand menu. The popularity of this piece has meant that it has been arranged for other instruments, and on mfiles you can find the Guitar Arrangement by Spanish composer Francisco Tárrega and an arrangement for Violin and Piano by Jim Paterson.
Here is a video of Chopin's Nocturne no.2 for piano:
The audio controls below allow you to play the music or download Chopin's Nocturne no.2 for piano as an MP3 file.
The image below is the first page of the sheet music, and here you can download the full sheet music for Chopin's Nocturne no.2 for piano in PDF format.