This piece, Hungarian Dance #1, was actually recorded on Edison Cylinder by Brahms himself in 1889 (YouTube below the fold). It’s a horrible recording, barely recognizable, and the voice is almost certainly not Brahms. It’s one of 21 Hungarian Dances of which the most famous is probably #5. They’re great favorites of piano teachers, this one is usually performed as a duet or a 4 hander.
In his capacity as director of the Moscow branch of the Russian Musical Society, Tchaikovsky invited Brahms, who was quite a celebrity, in as a guest conductor. March of the Slavs was composed in commemoration of a crushing defeat of the Serbian Army in 1876 by the Turks. It was written in 5 days and includes some familiar themes later used in his more well known 1812 Overture.
Brahms on Edison