If a spell is countered does it go to the graveyard or back to the owner's hand?

When a spell is countered, it is moved from the stack to its owner's graveyard as the default result. This is established by Comprehensive Rule 701.5a: 'To counter a spell or ability means to cancel it so it doesn't resolve and, if it's a spell, to put it into its owner's graveyard.'

The spell never resolves and its effects never happen. It simply ceases to exist on the stack and ends up in the graveyard, just like a permanent that is destroyed or a spell that has finished resolving.

Some specific cards deviate from this default. For example, Remand counters a spell but replaces the 'goes to graveyard' result by returning it to its owner's hand instead. This is a special exception written directly on the card, not the normal rule.

Concrete example: Your opponent casts Lightning Bolt targeting your creature. You cast Counterspell. Lightning Bolt is countered and goes to your opponent's graveyard — their creature does not take damage, and they do not get the card back in hand.

So unless the countering effect specifically says to return it to hand (or exile it, etc.), the graveyard is always the destination per CR 701.5a.