The Crown of Aragon was more or less a confederation. Each of the territories that constituted it had different legal systems, and the King had different attributions as King of Valencia, King of Naples, or Count of Barcelona. In fact, the King was crowned separately in each territory, as told by Ramon Muntaner in his chronicle, chapter XXIX:
Don Pedro went to Zaragoza, and there he gathered the Cortes, and was placed upon his head the crown of the Kingdom of Aragon very solemnly, and with great joy, and with great feasts. As he had been crowned in Aragon, he came to Valencia; in the same way the Cortes were great and he received the crown of the Kingdom of Valencia. After that, he went to Barcelona, where he gathered the Cortes. And there he received with great glory and joy the garland with which he was crowned Count of Barcelona and lord of all of Catalonia
More or less, the legal systems were mostly two: one based upon the Catalan constitutions, and another one based on the Fueros of Aragon. Mallorca had its institutions and legal system based on the Catalan constitutions, whileas Valencia had a hybrid between the Fueros and the Constitutions. The Fueros were charters granted by the King, whileas the Constitutions were pacted between the King (in his role as Count of Barcelona) and the Cortes. The King, as Count of Barcelona, was a primus inter pares amongst the Catalan counts, that is why he was not a king but a count with princely dignity, and he was never crowned but received a garland or laurel diadem.
Naples, as it was far away, was governed through a system of viceroys appointed by the King to act in his stead.