There are no challenges in the NHL. All replays are automatic: goals are reviewed by the video judge, with help from the war room in Toronto. Coaches do not have the ability to initiate replays in any form. Penalties are never reviewable, unless the timing relates to a goal. All goals are reviewed (at least cursorily) by the Video Goal Judge, and further review may be undertaken if he or the referees determine it necessary.
This is explained in the NHL Official Rules, where it lists the jobs of the Video Goal Judge (38.1):
(i) He will review replays of disputed goals when requested to do so by the Referees.
(ii) He will review replays of disputed goals when he observes an incident that was undetected by on-ice officials.
It later mentions that he reviews all goals:
38.2 Goals – Every goal is to be reviewed by the Video Goal Judge.
The review may not be in all that much detail, and may be short enough to not delay the game; it also may be in more detail and may slightly or significantly delay the game, if needed. This is somewhat covered in the rules.
Finally, it lists that the Video Goal Judge may also determine the correct game clock setting, which appears to be the only non-goal related act that is subject to replay.