Madrid, capital of Spain and home of the royal family. The city center with great architecture is amazingly small, with nice places to enjoy local food and wines and very walkable.
Seville is the capital of the Andalusian region that also includes Malaga and Granada. It is very Spanish here with historical buildings, Flamenco Music & Dance, horses, Paella and Sangria. Note that it can be really hot (40°C and more) in Summer...
Cordoba has a beautiful old city center with a lot of charm. The main attraction however is La Mesquita, a cathedral that started as a mosque. Cordoba is one of the hottest cities in Europa in Summer.
Granada is a major tourist destination in Andalusia because of the Alhambra. The nearby Sierra Nevada mountains (3000+m altitude) allow for skiing in winter, with a unique view of the Mediterranean.
Barcelona is capital of Catalonia and world-famous for soccer and architecture. You must visit the cathedral Sagrada Familia, designed by Gaudi, that has been under construction since 1883. Camp Nou, where FC Barcelona resides, is an experience of a different kind.
Málaga is an important tourist destination, known as "the capital of the Costa del Sol". An estimated 6 million tourists visit the city each year, who usually visit the birthplace of Pablo Picasso.
Segovia is located just north of Madrid and very well worth a day-trip. The main sights are the old city center with large Roman Aquaduct, the Cathedral and the Phantasy Castle by Alfonso II.
Cadiz is one of the oldest cities in Europe. It's location is quite unique, since it is surrounded almost completely by water (similar to Venice). James Bond used the city to resemble Cuba.