While the poem doesn't actually name the reindeer, it does describe them as "eight tiny reindeer," giving us the following list:
* Dasher
* Dancer
* Prancer
* Vixen
* Comet
* Cupid
* Donner
* Blitzen
The names themselves are not explicitly stated in the poem, but they have become so commonly associated with Santa's reindeer that they are now considered canonical.