In the world of animation and video games, character design is the craft of creating the characters that populate a given world.
This is an absolutely vital role. In cel animation (and 2D video games) the character designer is the person who, from the director's descriptions (unless the director is the character designer) draws the reference drawings that the animators use to draw each individual frame. How important is this? Basically, the character designer has the job of making a character that's supposed to be likeable actually look likeable; a girl that's supposed to be outstandingly beautiful has to actually BE outstandingly beautiful (as opposed to, say, a book, where you can just claim she is and let the reader's imagination do the rest.) In the world of animation, image is everything, and the character designs have to reflect both the physicality and the charisma of a given character.
I'm only familiar with Japanese character designers, but I well know how important they are in any animated endeavor.
Some well-known, talented character designers are Haruhiko Mikimoto, Kousuke Fujishima, Akemi Takada, Yoshitoshi ABe, and Yoshitaka Amano. (There are many more, both professional and amateur.)
Character design is becoming rapidly more important in video games as well. Nowadays, video game characters are mostly polygon models, but the digital modelers often need some kind of visual reference to work from. Remember those pen & watercolor renditions of the Metal Gear Solid characters that appeared in the manual? That was a wonderful example of character design. Looking at those images, it was easy to see what the polygon modelers had in mind when they were creating the characters. Ico is another marvel of 3D character design in video games. Just about any fighting game needs good character design in order to differentiate one badass kung-fu master from another. Pre-rendered computer imagery is no exception. I shouldn't even need to mention Pixar's characters in Toy Story or Monsters, Inc. -- rest assured that the modelers have excellent concept art to work from.
All in all, an underappreciated, but vital art.