The name of a young bird varies by species, so there is no truly unifying term to describe all young birds except in a generic sense. If this is the aim, such terms as nestling, fledgling, hatchling and chick can be applied.
For example, a baby dove is known as a squab or chick. A baby duck is known as a duckling. A baby eagle is known as a fledgling or eaglet. A baby goose is known as a gosling, and a baby hawk is known as an eyas. A baby swan is known as a cygnet or a flapper.