Bullfights are very popular spectacles in Spain, though perhaps nowadays the craze has died down a bit due to contrasting public opinions. Some see it as an art form, as it takes a skillful individual to avoid being gored and doing so with a certain degree of grace. Others view the controversial sport as an example of animal cruelty. Regardless of how you feel going into it, it's still good to experience it before placing a solid opinion on the matter, and around Spain you will certainly have your opportunity.
The Running of the Bulls is another interesting concept in Spain. The name pretty much says it all: a set of bulls are released down a path that is boarded off to ensure they travel down the established route. Meanwhile, a group of brave and / or crazy runners run in front of and alongside them to test how long they can last before they get turned into the bull's puppet or manage to get out of there. Not advisable to try yourself, but it can certainly be an interesting spectacle.
Bulls have been in Spanish culture for many years and their integration into the national traditions is still very important, especially to some older generations. As said, there are some who consider it in bad taste and it can actually depend on where you go in Spain. In Barcelona, for example, the government agreed to declare itself a non-bullfighting city which is a bold move but also a sign that bullfighting isn't as widely accepted as it once was. In other areas it still remains dominant, such as in Andalusia.