Listen to the original 1857 version of "Jingle Bells"

It was written in 1850 by James Lord Pierpont and published in 1857 as "The One Horse Open Sleigh." When it was republished in 1859 the title was changed to "Jingle Bells; or, The One Horse Open Sleigh."

There are many claims as to its original intention: a drinking song, a Sunday school choir song to be performed for Thanksgiving Day, or another version of a "sleighing song," which was a popular genre in those days. It may have been inspired by the sleigh races which were held in Medford, Massachusetts, where Pierpont composed the song. Its association with Christmas began in the 1860s and 1870s.

This is how it would have sounded back then. Note the major differences in the melody, slightly different wording in the chorus, and some completely different verses:
https://www.youtube.com/watch?v=3TV2JchT6ks

Over time the melody changed. By 1898 the more-familiar modern tune had replaced the original, as can be heard in this Edison cylinder recording from that year:
https://www.youtube.com/watch?v=pZoXLIrYru4

For further reading: https://en.wikipedia.org/wiki/Jingle_Bells