It is a Sunnah (action following the teachings of the Prophet) for
the time of slaughtering the 'Aqiqah for the newborn baby to be on the seventh day of his or her life. This is according to the Hadith of
Samurah ibn Jundub (may Allah be pleased with him) that the Messenger of Allah (peace be upon him) said:
A boy is in pledge for his `Aqiqah. It will be slaughtered for him on the seventh day, his head is to be shaved and he is to be named.
This was narrated by
Al-Tirmidhy and others.You mentioned what has been reported in a Hadith that 'Aqiqah is to be slaughtered on the seventh, fourteenth or twenty-first day. This is the Hadith of
Samurah saying that it should be on the seventh day. Specifying other times is the view of a group of Salaf (righteous predecessors).
Al-Tirmidhy (may Allah be merciful to him) said after the mentioned above Hadith of
Samurah: This is a Hadith Hasan Sahih (a Hadith that stands at a higher level than a mere Hasan Hadith, but at a bit lower level than Sahih). Acting upon this is endorsed by the scholars. They prefer that 'Aqiqah is to be slaughtered for the new baby boy on the seventh day. If it is not possible on the seventh day, it should be done on the fourteenth day. If it is not possible on that day, it should be done on the twenty-first day.The Sunnah (action following the teachings of the Prophet) is what I have learnt in the Hadith of
Samurah indicating the slaughter to take place on the seventh day. If the Waliy (guardian) fails to slaughter the animal on that day, the conditions of 'Aqiqah will be met because the objective is met by slaughtering 'Aqiqah at any time but he missed the commendable time.