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.
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.
San Sebastian is a superb beach location just west of the Pyrenees with excellent food (tapas). It should be on everyone's itinerary.
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.
Mallorca (aka Majorca) is a very well visited Spanish vacation island in the Mediterranean. Many Germans, Brits and other Europeans spend their vacation in one of the many resorts on one of the many beaches - or just in a Finca in the main land. Some even spend the whole winter there and enjoy the pleasant climate and atmosphere. Palma de Mallorca has excellent connections by air to many European cities.
Pamplona is best-known for its annual "running with the bulls"... Yes, these are real bulls and it is dangerous! Only the very sporty ones that have something to prove, should try this...
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...
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.