For me, character design is one of the hardest parts of my job as an illustrator. It's something that you'll come across in your career as an illustrator frequently. Few images are of landscapes or just products. Sooner or later you're going to have to enter the wonderful world of character design. The people or creatures in your pictures are after all the stars of your artwork. They deserve some time and attention to get them looking just right.
The first thing I do when I am designing any kind of character is to have a bit of a think about what it is I want to convey with this character. If it's an illustration for a fictional project, it's likely that the character will fall into one of two classical groups: a good guy, or a villain. It's important that your character conveys their personality. Unless it's done for comedy purposes, no-one is going to think a cute seal puppy with big eyes is capable of kidnapping the princess in the story.
Research note: Go to http://www.google.com and search for "disney villains". Look at each of the main villains from the Disney movies (there'll be lots of pictures of them on the sites you find listed). Disney were masters at making the bad guys look really evil and creepy. Note how they use the same little devices in each drawing: the high eye-brows, the long wicked smile, the hard, sharp edges and dark gloomy colors. Give your viewer as many visual clues as you can as to what your character is all about.
Another important part of character design, be it fiction or a factual educational project, is to find out as much as you can about the characters from the author / writer of the book or project you're illustrating. Educational projects will have much tighter boundaries normally about how the characters should look. If it's for a CD ROM then there may even be 'real life' actors playing the characters which will take most of the work out of your job. Read the manuscript very carefully also, and build up a profile on the character(s) you're going to be illustrating. Hair color, skin color, height, build are all important questions. Get to know your characters.
If you've got more freedom to build up the characters, or if this is one of your personal projects, then you have a lot more work to do in front of you. It's time to being the long (and often very difficult) process of building up your character from scratch. I always start off with a pencil and paper and just begin to do one drawing after another - often on scrap paper or the revers side of printouts I no longer need. These drawings should be fast and furious - a bit like what writers do when they're brainstorming. With each drawing you do, try something completely different with your character.
Variations
It's all about the variations. In one drawing give them short legs, in the next, try bigger eyes, in the next give them a moustache - that kind of thing. I usually get a feel when the character is looking right. Keep asking the question 'what if' - what if I give him a hat? What if I make the eyebrows bigger? What if I extend that mouth right across the length of his face?
Here's some sketches I did for a project which required a drawing of a menacing little spider:
As this was one of the main characters in the story, I made sure I got him looking just right. I must have done hundreds of sketches before settling on the one below:
Always keep your sketches too - they may prove useful for future illustration projects.