The Red Ribbon Army or the "World's Mightiest Evil Army" were once one of if not the most dangerous and feared criminal orginization on Earth. Not even the police or the army had been able to take them down. They became involved in the story of Dragon Ball after the head of the army--Commander Red-learned about the dragon balls and orginized a full scale search for them. The Army believed that they were searching for the balls so that Commander Red could wish for world domination. In actuality, he just wanted the balls so he could wish to be taller. However, Son Goku thwarted the army's plans and destroyed most of the army, with only the army's top secret scientist and android manufacturer, Dr. Gero, left.
Dr. Gero later developed androids strong enough to defeat Son Goku and launched an attack on him. However, thanks to the meddling of time traveller Trunks X, his plan did not work and the last trace of the army was annihalated.