From Muppet Wiki

Moe Cocker, a canine take-off of singer Joe Cocker, appeared on Sesame Street and performed "With a Little Yelp from My Friends," with a chorus of dogs. The song was a take-off of The Beatles' tune, later recorded by Cocker.

The same puppet has been used for various dogs on the series, including Watson and a dog in Elmo's World: Dogs.