Does ward trigger when a spell targets a creature or only when an activated ability does?

Ward is a triggered ability that watches for any spell or ability an opponent controls that targets the permanent with ward. This is stated explicitly in CR 702.164a: whenever a permanent with ward becomes the target of a spell or ability an opponent controls, that opponent must pay the ward cost or the spell or ability is countered.

So ward covers three broad categories: (1) spells — instants, sorceries, or any other spell that targets the creature; (2) activated abilities — such as a planeswalker's ability or an equipment's equip ability; and (3) triggered abilities that target the creature when controlled by an opponent. All three can cause ward to trigger.

One common misconception is that ward only stops targeted removal spells. In reality, if an opponent activates an ability like Viridian Longbow's tap ability and points it at your creature with ward, ward triggers just the same as it would for a Lightning Bolt.

Concrete example: Your creature has Ward {2}. Your opponent casts Lightning Bolt targeting it (a spell) — ward triggers and your opponent must pay {2} or the Bolt is countered. Later, your opponent uses Prodigal Sorcerer's tap ability to target the same creature — ward triggers again, and they must pay {2} or that ability is countered.

Note that ward does not trigger when the permanent is targeted by spells or abilities its controller or an ally controls — only opponents' spells and abilities matter (CR 702.164a).