What causes these white spots in/under my finger nails? Is it serious?

Leukonychia . Milk spots on nails are usually harmless & related to nail trauma (such as biting nails). Other causes can be Iron or protein deficiency, lead or arsenic poisoning, renal failure, cirrhosis, pneumonia, heart disease, psoriasis or marked psychological stress. Zinc or calcium deficiency are possible if on both finger & toe nails.
No need to worry. According to the american academy of dermatology, the white spots occur after injury to the nail. Fortunately, as the nail grows out, the spot will disappear. No need to worry.