The EU pet passport is the standard document for moving a pet between EU countries, including into Spain, and it's only issued to pet owners resident in the EU by an authorised vet in their home country. It records the microchip number and the full rabies vaccination history, and stays valid for the pet's life as long as the vaccination record is kept current.
If you live outside the EU, MAPA's guidance sets out a more involved process: you'll need an animal health certificate, signed by an official vet in the country of origin and presented at least in Spanish, along with a signed owner declaration and a certified copy of the pet's identification and vaccination records. If your country of origin isn't listed in Annex II of EU Regulation 577/2013 — broadly, the list of countries the EU treats as low-rabies-risk — your pet also needs a rabies antibody test at an approved laboratory, with a result of at least 0.5 IU/ml, and entry is only allowed starting three months after that blood sample was taken.
Whichever document applies, MAPA's guidance requires arriving through a designated Traveler's Point of Entry and declaring the pet to the Guardia Civil's customs post with its paperwork — a step that's specific to non-EU arrivals and worth building into your arrival time if it applies to you.