A Paramour is an unmarried lover of a noble. While many lords in Westeros keep illicit lovers called "mistresses" or lowborn "bedwarmers", "paramours" in the unique culture of Dorne are a publicly acknowledged and official relationship. Paramours are accorded a degree of formal social standing in Dorne, though this varies from relationship to relationship.
Some lords will legally marry one woman for political reasons, but maintain long term sexual and emotional relationships with their paramour. Such paramours may be kept private, or as an open secret, while others are quite public. It is not uncommon for some paramours to be considered a lord's real wife in all but name. Paramours in Dorne are a holdover from centuries ago when the Rhoynar ancestors of the Dornish used to live in city-states along the Rhoyne River in Essos, with a sophisticated courtly culture.
Dornish noblewomen may also have male paramours. Unlike the rest of the Seven Kingdoms, Dorne follows equal inheritance laws, and many women are politically powerful in their own right. Moreover, due to the more open sexual mores and flexibility of Dornish culture, it is not surprising for nobles to also have paramours of the same sex.
Sometimes however, lords and ladies are simply not in a social position to marry their beloved. For example, Prince Oberyn Martell of Sunspear is officially unmarried, but he fell in love with Ellaria Sand - an acknowledged bastard daughter of House Uller. Even in the relaxed values of Dorne, Oberyn could not wed Ellaria because it would be marrying beneath his station. Instead, Oberyn simply formally acknowledged Ellaria as his paramour and now cohabits with her, and she functionally lives as his wife. Ellaria has also had daughters by Oberyn, and though he has also formally acknowledged them, their parents are still not legally considered married so they must continue to use the bastard surname "Sand" (though unlike the rest of the Seven Kingdoms, bastardy is not considered very shameful in Dorne).
The other regions of the Seven Kingdoms find paramours disgraceful. Some may be diplomatic and extend them courtesy, recognizing that Dornish culture is different from the rest of Westeros, like Tywin Lannister did at the Wedding of Joffrey and Margaery when he was introduced to Ellaria by Oberyn. Others will openly look down on them as little more than prostitutes, though paramours are relationships based on love, not money.
Many marriages in Westeros are contracted not out of love but to cement political alliances, yet the Dornish think that people should be allowed to love whom they will. For their part, the Dornish think the other kingdoms are somewhat hypocritical: many lords in the rest of Westeros keep secret mistresses while trapped in loveless marriages, and the Dornish believe they are simply being honest by openly acknowledging these relationships.
It is not clear however if a Dornish lord or lady is allowed to have a paramour from an outside of their respective kingdom.