Muleta is the name of a stick the red cloth used in the final third (tercio de muleta or de muerte) of a bullfight hangs from. It is different from the cape used by the matador earlier in the fight (capote de brega).
The muleta obscures a sword, and as in his earlier work with the cape, bullfighter uses it to attract the bull in a series of passes, demonstrating his control over it. (Veterinary physiological aside: Bulls are actually dichromatic, so neither the cape nor the muleta need be red to serve its purpose. The red colour is just a matter of tradition.)
There are a number of distinct styles of pass, each with its own name. With the cape, for instance, the verónica is a pass in which the matador slowly swings the cape away from the charging bull while keeping his feet in the same position. The faena is the final series of passes before the kill, in which the matador uses the muleta to manoeuvre the bull into a position to stab it between the shoulders and through the heart. If this fails he must then cut the bull's spinal cord with a second sword, killing it instantly. The task of killing the bull is given to the matador alone; his title means literally "killer".