Read the script. You don't need to memorize it, but you do need to develop an understanding of what the film should look and feel like. Look at the character, prop and setting designs and learn how to draw them quickly from as many angles as possible and in as many positions as possible. The more accurate you can get your drawings to be, the better your storyboard will be. Take the time now to also research shot types and story progression. They will be invaluable when making your storyboard.
Consult with other members of the production team to ensure they have a say in what the storyboard looks like. Sit down with them and walk through the script, making sure that their vision is the same as yours.
Create your storyboard layout. On a piece of A4 paper, draw out 4 boxes with the same aspect ratio (the width:height ratio of the screen) as the animation the story board is for. The most common aspect ratio is 16:9, used for movies. Leave space below the boxes for any general notations, dialogue or notes on camera movement. Each sheet should have the scene name and page number on it so the layout can be organised properly.
Draft your storyboard. Work your way through the script, drafting out each shot in the boxes. Keep in mind that you are only drafting out your ideas. You can always go back and erase them. At this stage your characters do not need to be recognizable to anyone but you. This stage is for you to work out what you are going to do and should be done as quickly as possible. Listen to the voice track for the animation, if there is one, and make sure that your shot timing is correct.
Finalize the storyboard. Go over your rough work with harder and tighter lines. Make your characters recognizable and clear in detail. A storyboard does not need to be a flawless piece of art but, it should be communicate the scene as accurately as possible to the director and animators. The production team should know exactly what to do with a glance at your storyboard.