Snow is a supertype, not a card type. It has no rules meaning or function by itself, but spells and abilities may refer to it.

Some cards have additional effects for each {S} spent to cast them. You can cast these spells even if you don't spend any snow mana to cast them; their additional effects simply won't do anything.

The Kaldheim set doesn't have any cards with mana costs that include {S}, but some previous sets do. If an effect says such a spell costs {1} less to cast, that reduction doesn't apply to any {S} costs. This is also true for activated abilities that include {S} in their activation costs and effects that reduce those costs.

The {S} symbol is a generic mana symbol. It represents a cost that can be paid by one mana that was produced by a snow source. That mana can be any color or colorless.

Another snow Zombie creature you control will get only +1/+1 from Narfi's first ability.

Snow isn't a type of mana. If an effect says you may spend mana as though it were any type, you can't pay for {S} using mana that wasn't produced by a snow source.