To modern audiences the 'rock' aspect of the song can sound more country & western as it follows a style popular in the 50s that merges the two called 'rockabilly'. The single has been included on a number of Bobby's own albums, the most recent (and final) in 1983, and one in 1970 on an album itself titled Jingle Bell Rock.
I've already mentioned the many different cover versions of this song, but to those of us of around my age (28 at the time of typing), arguably the most well-known is the original largely due to its inclusion on the soundtrack of 1990's Christmas movie Home Alone.
Here's a vid:
And because it's Christmas and I'm feeling generous, here's Billy Idol's version, complete with promo-vid. Worth it for the facial expressions, I feel: