Showmanship is a judged presentation of livestock in hand. You'll see showmanship classes at 4-H shows, involving cattle, horses, goats and other animals. You'll see showmanship judged at dog and cat shows, too. Participants are judged on the clean and breed-appropriate appearance of the animal and themselves and the quality of their formal presentation, such as standing the animal nicely, walking or jogging the animal for the judge and so forth.
In the horse world, particularly with popular "western" breeds such as Paints, Quarter Horses and Arabians, showmanship competitions have gotten quite fancy. Handlers dress up in fancy attire, and the horses are presented in complex patterns, that include walking, jogging, halting, pivots and circles. You can read more about the details of horse showmanship here and here.
Take a look at this video for an example of the type of patterns and presentation required: