The description below is taken from the Wikipedia entry on bullfighting to avoid issues of copyright and partiality (although, it must be said I have contributed to the article myself and my writings elsewhere have been used as sources by other editors). To really understand, one needs to see it, or at the least read a vivid description of a fight after reading the structure below. I attempted one such vivid description in my Prospect magazine essay the link to which is on the right hand toolbar.
Spanish-style bullfighting is called la corrida de toros (“the running of bulls”) or la fiesta brava (“the ferocious festival”). In a traditional corrida, three matadores (“killers”), each fight two bulls, each of which is at least four years old and weighs 460–600 kg. Each matador has six assistants — two picadores (“lancers”) mounted on horseback, three banderilleros (“flagmen”) – who along with the matadors are collectively known as toreros (“bullfighters”) – and a mozo de espada (“sword page”). Collectively they comprise a cuadrilla (“entourage”).
The modern corrida is highly ritualized, with three distinct stages or tercios (“thirds”), the start of each being announced by a trumpet sound. The participants first enter the arena in a parade, called the paseíllo, to salute the presiding dignitary, accompanied by band music. Torero costumes are inspired by 18th century Andalusian clothing, and matadors are easily distinguished by the gold of their traje de luces (“suit of lights”) as opposed to the lesser banderilleros who are also called toreros de plata (“bullfighters of silver”).
Next, the bull enters the ring to be tested for ferocity by the matador and banderilleros with the magenta and gold capote (“dress cape”). This is the first stage, the tercio de varas (“the lancing third”), and the matador first confronts the bull with the capote, observing the behaviour of the bull while performing a tanda (“series of passes”) to impress the crowd.
Next, a picador enters the arena on horseback armed with a vara (“lance”). To protect the horse from the bull’s horns, the horse is surrounded by a peto — a protective mattress-like covering. Prior to 1930, the horse did not wear any protection, and the bull would usually disembowel the horse during this stage. Until this change was instituted, the number of horses killed during a fight was higher than the number of bulls killed. At this point, the picador stabs just behind the morillo, a mound of muscle on the fighting bull’s neck, weakening the neck muscles and leading to the animal’s first loss of blood. The manner in which the bull charges the horse provides important clues to the matador about which side the bull favors. If the picador is successful, the bull will hold its head and horns lower during the following stages of the fight. This makes the bull’s charges less dangerous and more reliable, enabling the matador to perform.
In the next stage, the tercio de banderillas (“the third of flags”), the three banderilleros each attempt to plant two banderillas, sharp barbed sticks into the bull’s shoudlers. These anger and invigorate the bull who has been tired by his attacks on the horse and the damage he has taken from the lance. Sometimes a matador will place his own banderillas.
In the final stage, the tercio de muerte (“the third of death”), the matador re-enters the ring alone with a small red cape, or muleta, and a sword. It is a common misconception that the color red is supposed to anger the bull, but bulls, in fact, are colorblind. The cape is thought to be red to mask the bull’s blood, although this is now also a matter of tradition. The matador uses his cape to attract the bull in a series of passes which serve the dual purpose of wearing the animal down for kill and producing a beautiful display or faena. He may also demonstrate his domination over the bull by caping it especially close to his bod. The faena is the entire performance with the muleta and it is usually broken down into tandas, “series”, of passes. The faena ends with a final series of passes in which the matador with a muleta attempts to maneuver the bull into a position to stab it between the shoulder blades and through the aorta or heart. The act of thrusting the sword is called an estocada.
If the matador has performed particularly well, the crowd may petition the president to award the matador an ear of the bull by waving white handkerchiefs. If his performance was exceptional, he will award two, and in certain more rural rings a tail can still be awarded. Very rarely, if the public or the matador believe that the bull has fought bravely, they may petition the president of the plaza to grant the bull an indulto before the tercio de muerte. This is when the bull’s life is spared and allowed to leave the ring alive and return to the ranch where it came from. Then the bull becomes a stud bull for the rest of his life.