Since in Essos, particularly in the Free Cities, the city-state form of government has prevailed, the title "Prince" has been widely used. Before the Rhoynar moved to Dorne, they also lived in city-states so their rulers were styled Prince or Princess. The title Prince of Pentos is still used. 
In Westeros, the title is usually conferred to the children of the king. Prior to the War of the Usurper, the heir to the king on the Iron Throne held the title "Prince of Dragonstone",  and one his other sons held the title "Prince of Summerhall". The rest of the children of the Monarch were styled Prince or Princess. Ever since the deposition of the Targaryens during Robert's Rebellion, the title "Prince of Dragonstone" has not been used; the children of the monarch are referred to only as Prince or Princess.
In Dorne, the ruler of that "kingdom" is styled Prince, following the Rhoynar tradition. His sons and daughters are also styled "Prince" or "Princess of Dorne," as we can see with Oberyn Martell and the children of Prince Doran Martell.