In Japan, "ki" has many meanings. One of the ones the anime series like to focus on is its psychic-power connotation.
I personally know for myself that there is something to that connotation; however, all my experience has pointed me to the direction not of power, but of intention and frame of mind. In other words, you can go off on the side track of power, but it is a red herring, an eventual dead-end if your ultimate goal is that power. The true path is within: within one's heart and mind, and the real battle (IMHO) is the one that good anime shows hint at on occasion: the battle for internal discipline and a compassionate spirit (a battle that never ends, by the way).
In aikido, especially Ki Society aikido, I believe that "ki" means a specific frame of mind that is relaxed, calm, centered, aware, compassionate, and benevolent. The results of this are sometimes "feats" that may appear magical, but they are the automatic results of having that mindset. Is there psychic power involved? Not in the sense of the blasting away that's in anime, but in the sense of the real power of true calmness and peacefulness in the face of confusion.
I am reminded again of the story of "real aikido," which is not to use a martial art to quell violence, but to prevent violence AND bring healing by extending compassion to troubled souls.
P.S. One of the simplest and best examples I know of Real Ki is the extension of good will that is an honest smile or a small expression of caring. Just think -- one smile, or a few kind words, can positively change the day for another person, and possibly even cause a chain reaction of good will, like a "ripple effect." Now, for something so small to affect so much -- isn't that true power?