What is a Certificate of Airworthiness for? How do I get it?

An Airworthiness Certificate is the document that accredits the aircraft to which it refers, as being in a position to fly by meeting certain safety requirements. 
AESA would be the authority responsible for issuing Airworthiness Certificates for aircraft that are or will be registered in Spain. 

To apply for the Airworthiness Certificate there is a user manual for EASA aircraft and a user manual for NO EASA aircraft (Excluded construction aircraft by amateurs and experimentals who have their own guide). This manual also indicates how to carry out subsequent activities after obtaining the Certificate of Airworthiness, such as modifications, suspensions, etc.