The goal for wu wei is to get out of your own way, so to speak. This is like when you are playing an instrument and if you start thinking about playing the instrument, then you will get in your own way and interfere with your own playing. It is aimless action, because if there was a goal that you need to aim at and hit, then you will develop anxiety about this goal. Zhuangzi made a point of this, where he writes about an archer who at first didn't have anything to aim at. When there was nothing to aim at, the archer was happy and content with his being. He was practicing wu wei. But, then he set up a target and "got in his own way." He was going against the Tao and the natural course of things by having to hit that goal.
A dramatic description of wu wei is found in chapter 2 of Zhuang Zi:
A fully achieved person is like a spirit! The great marshes could be set on fire, but she wouldn't feel hot. The rivers in China could all freeze over, but she wouldn't feel cold. Thunder could suddenly echo through the mountains, wind could cause a tsunami in the ocean, but she wouldn't be startled. A person like that could ride through the sky on the floating clouds, straddle the sun and moon, and travel beyond the four seas. Neither death nor life can cause changes within her, and there's little reason for her to even consider benefit or harm.
This passage is metaphorical. To a Taoist, things arise dependently. The soul and body go together, because if there were no soul, there would be no body and if there were no body, there would be no soul. All these arise dependently like this (this is the meaning of the Yin-Yang symbol; if there were no yin, there would be no yang and if there were no yang, there would be no yin). A person who follows the principle of wu wei thus realizes how ridiculous it is to cling to good and to obsessively stay away from evil. By realizing how things arise dependently, a Taoist is able to accept the bad and is then able to have no goal to aim at. When Zhuangzi is saying a fully achieved person is like a spirit, he is saying that a fully achieved person has accepted the bad things in life and does not strive to get away from them and instead accepts them.