Does ward trigger when a spell or ability targets the creature or only when an opponent targets it?

Ward is a triggered ability that triggers whenever the permanent with ward becomes the target of a spell or ability an opponent controls (CR 702.164a). This means it does not trigger if you target your own creature, or if a teammate targets it in a multiplayer game where you share a team.

The trigger covers any spell or ability controlled by an opponent — not just creature spells. A removal spell like Murder, a targeting activated ability like that of Prodigal Sorcerer, or an opponent's aura being cast at your creature will all trigger ward equally.

When ward triggers, the opponent who controls the spell or ability must pay the ward cost (e.g., pay 2 life, discard a card, pay {2}). If they don't or can't pay, the spell or ability is countered (CR 702.164b). Note that ward doesn't stop the spell from being cast — it only counters it after the fact if the cost isn't paid.

Concrete example: You control Reidane, God of the Worthy (Ward {2}). Your opponent casts Shock targeting it. Ward triggers. Your opponent must pay an additional {2} or Shock is countered. However, if you somehow targeted your own Reidane with a spell, ward would not trigger at all.

Also note: ward only cares about targeting. Non-targeted effects — like a spell that destroys all creatures — completely bypass ward (CR 702.164a).