In terms of physical activity, running was an ability to move faster than one's typical gait, and exerted more energy than one's normal speed. The faster pace generated by a humanoid body's legs was useful for combat situations, or simply exercise or recreation. Time was also considered to be able to "run", as well as computer systems and programs, scans, among other things.
During her youth, Uhura used to run the hundred meter dash in record time. She found that she was not as fast in her later years, when she attempted to escape from a Kzinti police web in 2269 and was recaptured. (TAS: "The Slaver Weapon")
In 2368, Captain Jean-Luc Picard met again with his mentor during his days at Starfleet Academy, Boothby. Boothby recalled when Picard beat a Ligonian in a wrestling match in fourteen seconds with a reverse body lift. Picard remarked that it was all in his legs, with the running he used to do. (TNG: "The First Duty")
The Starfleet Academy marathon was an example of an organized sporting event involving running. Academy instructors also ordered cadets to run laps as punishment for misdemeanours or simply as a method of officer training. (TNG: "The Best of Both Worlds, Part II"; VOY: "Learning Curve")