An Apostille is a kind of attestation that legalizes documents in a specific format acceptable to all countries belonging to the Hague Convention. In essence, it is an international attestation accepted in around 105 nations including most of the nations in the West.
Attestation is done for all the nations that are not a member of the Hague Convention and do not accept Apostille.
Under usual circumstances, a document does not require attestation from the concerned Embassy once it is apostilled.