The song was written by Johnny Marks and became incredibly popular, making Rudolph a beloved Christmas icon.
What is the song with all names of reindeer?
The song you're thinking of is "Rudolph the Red-Nosed Reindeer". While the song itself doesn't list all the reindeer names, the lyrics do mention Rudolph, Dasher, Dancer, Prancer, Vixen, Comet, Cupid, Donner, and Blitzen.