What happens to an Aura spell that is countered — does it go to the graveyard or stay on the stack?

When a spell is countered, it is moved from the stack to its owner's graveyard. This applies to all spell types, including Aura spells. The stack entry simply ceases to exist as a spell and the card ends up in the graveyard (CR 701.5a).

An Aura only becomes an Aura permanent — and attaches to something — when it resolves successfully. While it is still on the stack, it is simply an Aura spell targeting an object or player. Countering it prevents resolution entirely, so the attachment step never happens (CR 303.4).

One nuance: if the target of the Aura spell becomes illegal before it resolves (e.g., the target gains hexproof or leaves the battlefield), the spell is countered by game rules under CR 608.2b, and the Aura card still goes to the graveyard — not back to the battlefield or anywhere else.

Example: You cast Pacifism targeting your opponent's creature. Your opponent responds with Counterspell. Pacifism is countered and goes to your graveyard; the creature remains unaffected and free to act.

Summary: regardless of why the Aura spell was countered (by another spell/ability or by game rules), the card always moves from the stack to the graveyard (CR 701.5a, 608.2b).