It was the custom of the time of Reconquista in the Iberian peninsula for a king to put among his titles the places he had conquered. So D. Alfonso III of Portugal, once he had conquered fully Algarve (Southern Portugal), styled himself D. Alfonso III, king of Portugal and Algarve.

This title remained in use afterwards because the emir of Algarve fled to Castille as Portugal was conquering his territories. He then swore fealty to the Castillian king, Alfonso X, and Alfonso X of Castille started puting among his titles that he also was king of Algarve.

So we had two claims to the Algarve region. One, the Portuguese, by right of holy (re)conquest, and the other, the Castillian, by right of fealty. The Portuguese didn't recognize the Castillian claim and always put in text and in formal terms that they were the rightful kings of Algarve, not the Castillians. That's why the term stuck for longer than a generation.

Portugal and Castille would at first settle this question in the treaty of Badajoz in 1248, with Alfonso X promising the title of King of Algarve to his nephew, the Portuguese Dinis. But the succession after Dinis I was kind of complicated, and one of Dinis' sons went to Castille trying to find support to topple his brother. Afterwards, the Castillian kings would keep on using the term "king of Algarve" at the same time as the Portuguese kings were doing so. So the Portuguese also kept on using it, to stress that they were the rightful kings of that region. D. isabel I of Spain was the first monarch to ascend to the Spanish throne, in 1831, and not use the title "Queen of Algarve". But the Spanish would go back to it, and the current Spanish king, Felipe VI, has, among his many titles, king of the Algarves.