The earliest likely human voice appeared on a Phonautograph, in 1857 by Edouard Leon Scott de Martinville. The oldest ones correctable to our ears and unambiguously human voices are his, from April 1860; see the first entry here. However, these Phonautograph "recordings" were not originally meant for playback--rather they seem to have been meant to measure acoustics via movement of a stiff fiber on paper coated with soot or lampblack, and there was no way to play them back then in any case. They only became readable to souund via very modern technology and through the fortuitous introduction of a tuning fork (which provided a standard to correct against) to the 1860 sessions, not because they were intentionally designed for listening purposes.

The oldest deliberate recording for playback available may be the tin foil experimental cylinder from one of Edison's phonograph demonstrations in 1878. That was the year that he received the patent on the phonograph (Feb. 19th, 1878) but this demonstration was on June 22nd, 1878, in St. Lois. According to Edison's own notes, the principle existed in 1877, but intelligible playback only came in December of that year. We do not seem to have any of those very earliest attempts; all of the "1877" claims I've seen around are actually the 1878 recording, or something else. Given the fragility of tin foil, it is not wholly surprising that there are few really early recordings. It's also possible that damaged test foils were re-processed for new tests.

There is another candidate, however, in the experimental device from Frank Lambert, who was trying to apply Edison's device to the creation of a talking clock using a more durable lead cylinder. I don't see a specific month and date attached to the prototype beyond "1878," so it's hard to know which is really the oldest, but Lambert's is surely the oldest one still playable in its original form because tinfoil was so exceedingly fragile. Still, it can be hard to decipher, and audiophiles so inclined have spent some time doing so for at least a decade. So Edison's cylinders clearly came first, but Lambert's may be the oldest original one extant. If anyone has a good, clear date for Lambert's cylinder, I'd love to know.

(Interesting side note: there's a large library online via UCSB that offers many recordings from wax cylinders, but none there are the oldest--they point to the site I linked above for the Phonautograph. Still, it's quite a collection.)

[edit: spelling; fixed a link; noted importance of tuning fork]