Does a spell fizzle if all of its targets become illegal before it resolves?

When a spell is cast, its targets are chosen and locked in. Before that spell resolves, players may respond with other spells and abilities. If every single target chosen for the spell becomes illegal by the time it would resolve — because the target was destroyed, bounced, gained hexproof, became an invalid target type, etc. — the spell is countered by the rules of the game. This is informally called "fizzling." (CR 608.2b)

A target is illegal if it no longer exists on the battlefield (or in the appropriate zone), no longer meets the targeting requirements written on the spell, or has gained protection or hexproof that prevents it from being targeted. (CR 608.2b, 115.4)

Crucially, this only causes the spell to be countered if all targets are illegal. If a spell has multiple targets and at least one target is still legal, the spell still resolves, but it only affects the legal targets. Any parts of the spell that referred to illegal targets are simply ignored. (CR 608.2b)

Concrete example: You cast Lightning Bolt targeting your opponent's 1/1 creature. In response, your opponent casts Giant Growth on that creature... wait, that doesn't make it illegal. Instead, your opponent casts Unsummon, returning the creature to their hand. Now Lightning Bolt has no legal target, so it is countered by the rules and goes to the graveyard without resolving. Your opponent's creature is safe in hand.

Note that spells countered this way still go to the graveyard (or the appropriate zone), and any "when you cast" or "whenever this is countered" triggers will still function normally. The spell simply never resolves and produces no effect. (CR 608.2b)