Composer: Johann Sebastian Bach;
Bach wrote Air on the G String in the early 1700s as the second movement of his Orchestral Suite in D major. It wasn't given the "G string" title though until the late 19th century when violinist August Wilhelmj discovered that if he transposed the melody down a 9th (one note more than an octave), it could be performed completely on the violin's G string. (So, no - the G string idea is probably not what you might have first thought!)
It is a beautiful, quietly majestic tune that is often chosen for weddings, as it can be performed by a variety of instruments and instrumental combinations. I've included some free sheet music below: an arrangement for piano and another for string quartet.