Henry was a puppy.

In 2369, Henry was sitting on the couch in Amanda Rogers's quarters aboard the USS Enterprise-D while she was talking to Q. (TNG: "True Q")

Henry, deleted scene

Henry in the deleted scene

Henry was played by an unknown animal performer.
The name "Henry" appears in the script of "True Q", though the scene which the name was used in was cut from the aired version (even though the dog itself still appears in the final version of the episode).
According to the script, Henry was owned by Ensign Janklow. Janklow was scheduled to go to the Gantol excavations for three weeks, and was looking for someone to take care of Henry. Counselor Deanna Troi approached Amanda Rogers, who quickly agreed. While no mention of Henry was made during the final version of the episode, he can be seen sleeping on a couch behind Q shortly before he and Amanda begin to chase each other around the ship.