Free mechanic

The free mechanic is an ability which untaps a number of lands equal to a spell's converted mana cost. For creature spells, it is an "enters the battlefield" triggered ability. For non-creature spells, it is part of the spells' rules text. Spells using the free mechanic were originally printed in Urza's block and were all blue. The mechanic made a cameo years later in Dominaria. Zacama, Primal Calamity, printed in Rivals of Ixalan also uses the mechanic, but has a safety valve of only triggering if the creature was cast. The mechanic proved to be fundamentally broken.^[1]

Rulings

You don't choose which lands to untap until the spell or triggered ability resolves.
You may untap any lands. This includes lands you didn't tap to produce mana used to cast the spell, lands you don't control, and lands that have shroud or otherwise can't be targeted by spells or abilities.
If a spell with the free mechanic is countered, none of its effects happen. You won't get to untap any lands.
If a permanent with the free mechanic enters the battlefield ability enters the battlefield under your control as the result of a spell or ability, its "enters the battlefield" ability will still trigger even though it wasn't cast. You'll get to untap lands when the ability resolves.

Great Whale — 5/5 Whale creature.
Peregrine Drake — 2/3 Drake creature with flying.
Rewind — an instant with "counter target spell."
Time Spiral — a sorcery that makes all players shuffle their hands and graveyards into their libraries and draw seven cards, and is exiled afterward.

Cloud of Faeries — a 1/1 Faerie creature with flying.
Frantic Search — an instant with "Draw two cards, then choose and discard two cards."
Palinchron — 4/5 Illusion creature with flying and ": Return Palinchron to owner's hand."
Snap — an instant with "Return target creature to owner's hand."