Nonartifact spells cost {1} more to cast.
The ability affects each spell that's not an artifact spell, including your own.
The ability affects the total cost of each nonartifact spell, but it doesn't change that spell's mana cost or mana value. The extra mana also doesn't change how many times you kicked the spell or what the value of X is (if applicable).
When determining a spell's total cost, effects that increase the cost are applied before effects that reduce the cost.