Biography
Paternal instinct can transform a Bumbling Dad into an Action Dad. If someone threatens his kids they will soon wish they'd never come within a mile of them. This is because fathers are expected to take care of their family and this naturally extends to keeping them safe. Such occasions serve as a way for a father to prove his worthiness—see A Real Man Is a Killer. Expect his children to have a newfound respect for their father and for them to brag that My Dad Can Beat Up Your Dad. If their relationship was previously strained expect it to improve.
Often Papa Wolf incidents serve as a way to reveal that a Non-Action Guy is really a Retired Badass or a Crouching Moron, Hidden Badass (or even a Retired Monster). In contrast to an Overprotective Dad (who sometimes thinks he's this; there is no overlap between them), a Papa Wolf is always portrayed heroically because the latter is defending their kids from genuine threats instead of imagined ones.
A Team Dad may display a streak of this, but the Papa is more likely to be related to his children by blood or through formal adoption, and the children tend to be younger, which may be part of why the Papa Wolf is more oriented toward protecting them rather than training them to defend themselves. However even completely grown children can summon this response in the face of crisis because they are still his children no matter how old or strong they get.
This is the Spear Counterpart to Mama Bear. When Mama Bear and Papa Wolf team up, no force on earth can stop them.
Subtrope of Beware the Nice Ones. See also A Father to His Men and Family Man. Combining this with Disproportionate Retribution can lead to a Knight Templar Parent. If the guy is a teacher instead, he's a Badass Teacher. If the guy doing this is a sibling/cousin, you get Big Brother Instinct. Inversely, see Parents in Distress for the kids rescuing the dad. Evil characters can use this too; after all, Even Evil Has Loved Ones. A subtrope of the Papa Wolf is the Badass and Child Duo, where an adult male badass takes it upon himself to protect an orphaned, unrelated young child. See also Cub Cues Protective Parent for examples from the animal kingdom, which might include a literal wolf.
Remember when adding examples that this is Always Male. The female equivalent is Mama Bear, so all Distaff Counterparts should be placed there. When Mama Bear and Papa Wolf team up, it's a Battle Couple and all pairs should be placed there. Parents in Distress is the inversion, when Papa needs to be bailed out by the kids, and Extremely Protective Child is when the child exhibits this kind of general protectiveness over one or both of their parents.
Also, note that this is not the trope for being protective of one's friends, unless of course it is something like an Intergenerational Friendship. In that case, it is okay. Otherwise, don't do it. Tropes about helping friends should go to A Friend in Need, The Power of Friendship etc.