What happens when a spell that counters target spell targets a spell with split second?

Split second (CR 702.61) means that as long as a spell with split second is on the stack, players cannot cast other spells or activate abilities that are not mana abilities. This effectively 'locks' the stack while that spell is resolving its way to the top.

Because a counter spell is itself a spell, casting it in response to a split-second spell is illegal — you simply cannot cast it at that point (CR 702.61b). The split-second spell will resolve without the counter spell ever being cast in response.

Note that split second does not stop triggered abilities from triggering or going on the stack, and it does not stop players from activating mana abilities (CR 702.61b). It only prevents new spells from being cast and non-mana activated abilities from being activated while the split-second spell is on the stack.

Concrete example: Your opponent casts Sudden Death (which has split second) targeting your creature. You want to cast Counterspell to counter it — but you cannot. Split second prevents you from casting Counterspell while Sudden Death is on the stack, so Sudden Death resolves and your creature gets -4/-4.

There is one edge case: if somehow a counter spell were already on the stack before the split-second spell was cast, and the counter spell could target spells cast after it (unusual templating), that would be a separate analysis — but in standard play, you cannot cast a counter spell in response to a split-second spell.