Considering its name means "Fortress of the Christian Kings," it's quite ironic that the Alcazar de los Reyes Cristianos was designed by Mudejars (Muslims who remained in formerly Muslim areas of the Iberian Peninsula that had been reconquered by Catholic forces) in the style that bears their name. Although the present Mudejar structure was completed in 1328 on the orders of King Alfonso XI of Castile, it rests quite literally on far more ancient foundations, including the city's Roman walls and fortifications built by the Visigoths and Moors. Architectural niceties aside, the Alcazar's name is singularly appropriate on the basis of the role it subsequently came to serve.
Cordoba's historic importance as a center of intellectual and commercial life within the Iberian Peninsula led to the monarchs of Castile holding court in the Alcazar's palace with increasing regularity. It was also strategically positioned for military campaigns against the Moors, particularly in the 15th century, after their presence in Iberia had been reduced to the city-state of Granada. In 1485, Queen Isabella of Castile and Leon (whose marriage to Ferdinand II of Aragon created Spain as we now know it) moved her court there permanently in order to better supervise the campaign against Granada. Consequently, it was there that in 1492 that Ferdinand and Isabella gave their assent and patronage to his voyage across the Atlantic. Although Spain's court subsequently moved, it left behind a grim reminder of its presence in the Alcázar in the form the Inquisition, first instituted in 1481, which imprisoned, tortured, and killed suspected heretics within the fortress' walls until well into the 19th century.
Unfortunately, there is no descriptive information inside the Alcazar itself to provide visitors with any sense of this history, as it's treated more as a municipal edifice than a tourist attraction. A few Roman mosaics and statues found nearby, a plaque honoring the Guancha queen who was briefly imprisoned within the complex, and three gigantic statutes honoring Ferdinand, Isabella, and Columbus in the complex gardens are all that allude to the momentous events that transpired within it. Indeed, it was only after visiting that I found that part of the battlements visitors are permitted to climb (although only 15 at a time), known as the Inquisition Tower!
There's some disagreement over whether the gardens themselves, Cordoba's most elaborately manicured but not as pleasant as those which surround the city, date to Spanish or Moorish rule, although such fine points of history tend to get lost among the hordes you're certain to find inside them unless you make a point of visiting just as they open in the morning! Once inside, it's also worth exploring the Mudajar baths that adjoin them. Despite these points of interest, even as a history-lover I can't recommend paying the €3 entry fee, but it's worth a look if you've already purchased the Cordoba Card or are visiting on a Friday, when it's free (and consequently extremely crowded.)