What happens to an Aura on the stack if the target permanent is destroyed before the Aura resolves?

When a spell or ability resolves, the game first checks whether all of its targets are still legal. If every target of a spell is illegal at the time of resolution, the spell is countered by the game rules (not by any player) and placed into its owner's graveyard without resolving. This is sometimes called 'fizzling.' (CR 608.2b)

An Aura spell on the stack must target a permanent (or player, depending on what it can enchant) as it is being cast, per CR 303.4. If that target permanent is destroyed, bounced, or otherwise leaves the battlefield before the Aura resolves, the target becomes illegal. Since an Aura spell has only one target, losing that one target means all targets are illegal, so the Aura is countered by the game rules and goes to its owner's graveyard. (CR 608.2b)

Importantly, the Aura never actually enters the battlefield in this scenario, so no 'enters the battlefield' triggers from the Aura will occur, and no enchantment-related effects will apply.

Concrete example: You cast Pacifism targeting your opponent's Grizzly Bears. In response, your opponent casts Giant Growth and then sacrifices the Bears to an effect, moving them to the graveyard. When Pacifism tries to resolve, its target is no longer on the battlefield and is therefore illegal. Pacifism is countered by the game rules and placed into your graveyard.

Note: If the permanent is given hexproof or protection after the Aura is cast but before it resolves, the same fizzle outcome applies, since the target would again be illegal at resolution time per CR 608.2b and CR 115.7b.