Auld Lang Syne is a traditional song sung for New Year's Eve
Here is a version of Auld Lang Syne, sung by CelticKev on youtube. Celtic Kev states he is performing a version that Robert Burns himself would have performed in his own time, up until he died 21 July 21, 1796.
Songs evolve and Robert Burns borrowed many tunes for his poems, and even adapted other folk songs.
The ballad “Old Long Syne” printed in 1711 by James Watson. Burns was born in January 25, 1759 – a full 48 years after Watson printed his ballad.
In 1788, Robert Burns composed his version of “Auld Lang Syne” based on words “took down from an old man” and set to the tune of a traditional folk song (Roud # 6294), when Burns would have been 29 years old.
Wikipedia Auld_Lang_Syne writes”
Robert Burns sent a copy of the original song to the Scots Musical Museum with the remark, “The following song, an old song, of the olden times, and which has never been in print, nor even in manuscript until I took it down from an old man”.
Some of the lyrics were indeed “collected” rather than composed by the
poet; the ballad “Old Long Syne” printed in 1711 by James Watson shows
considerable similarity in the first verse and the chorus to Burns'
and is almost certainly derived from the same “old song”. It is a fair
supposition to attribute the rest of the poem to Burns himself.
There is some doubt as to whether the melody used today is the same
one Burns originally intended, but it is widely used in Scotland and in
the rest of the world.
In 1855, different words were written for the Auld Lang Syne tune by
Albert Laighton and titled, “Song of the Old Folks.” This song was
included in the tunebook, Father Kemp's Old Folks Concert Tunes published in Boston, Massachusetts in 1860. 
Singing the song on Hogmanay or New Year's Eve very quickly became a Scots custom
that soon spread to other parts of the British Isles. As Scots (and
other Britons) emigrated around the world, they took the song with them.
Canadian band leader Guy Lombardo
is often credited with popularising the use of the song at New Year’s
celebrations in America, through his annual broadcasts on radio and
television, beginning in 1929. The song became his trademark. In
addition to his live broadcasts, Lombardo recorded the song more than
once. His first recording was in 1939. A later recording on 29 September
1947 was issued as a single by Decca Records as catalog #24260.
See Auld Lang Syne on Wikipedia: