In humans, neonatal means "having to do with the first four weeks of life (after
birth)". (Most dictionaries define neonatal as having to do with the first month of life, but my medical dictionary is more specific.)
From neo-, meaning new, and natus, meaning born.
Update -- for cats (kittens), the neonatal period lasts for only two weeks.
For dogs it's from birth to 12 Days. (Sometimes given as birth to two weeks).