Hushwing Gryff’s ability stops a creature’s own enters-the-battlefield triggered abilities as well as other triggered abilities that would trigger when a creature enters the battlefield.

Abilities that apply “as [this creature] enters the battlefield,” such as choosing a creature to copy with Mercurial Pretender, are also unaffected.

The trigger event doesn’t have to specify “creatures” entering the battlefield. For example, Amulet of Vigor says “Whenever a permanent enters the battlefield tapped and under your control, untap it.” If a creature enters the battlefield tapped and under your control, Amulet of Vigor’s ability would not trigger. If a land (that isn’t also a creature) enters the battlefield tapped and under your control, Amulet of Vigor’s ability would trigger.

If Hushwing Gryff and another creature enter the battlefield at the same time, neither creature entering the battlefield will cause triggered abilities to trigger.

Look at the permanent as it exists on the battlefield, taking into account continuous effects, to determine whether any triggered abilities will trigger. For example, if you control March of the Machines, which says, in part, “Each noncreature artifact is an artifact creature,” each artifact will be a creature at the time it enters the battlefield and will not cause triggered abilities to trigger.

Triggered abilities use the word “when,” “whenever,” or “at.” They’re often written as “[Trigger condition], [effect].”

Abilities that create replacement effects, such as a permanent entering the battlefield tapped or with counters on it, are unaffected.