A good story is a good story, no matter the medium. When you move to a visual and auditory medium such as animation, you have a much wider field to play with, allowing you to show things you could only normally describe without limiting yourself because of character point of view or other boundaries of the medium. Too, there are drawbacks, such as lacking much room for inner monologue in an animated scene, and the need to show visually a character's thought process without narrating all of it. So what does good storytelling encompass in an animation?
First of all, planning.
When writing a book, some people outline, and some people pants it. Pantsing really doesn't work in animation. The rare genius may be able to pull it off, but for the most part you have to plan ahead because having to go back to edit is much more time-consuming than editing a book, and mistakes can be much more difficult to fix. Learn how to write an outline, then turn that outline into a storyboard, and expand that storyboard into a good, tight script. Understand your characters - their motivations, and their goals. Understand your plot and the effect you want to have on your audience.
Pacing in storytelling is key. Too fast, and people get lost. Too slow, and people get bored and drift off. Pacing isn't about just finding the right speed at which to move your story along, though. It's about variation, knowing when to animate a high-intensity scene and when to stop to linger on a poignant moment, and how to time the differently-paced scenes so that they come together with breathless perfection that grips the audience and never lets them go.
Tension, anticipation, and the three-act storyline.
Your classic storyline is broken into three acts: the setup, the execution, and the finale. The setup gives us the goal of the story and introduces our main characters, with a hook that leads us into the execution: the journey that takes us to the final conflict. Then the final conflict resolves the plot and achieves the goal established in the first act, while accomplishing the climax of the character development arc that should have been taking place throughout the entire story. This is as true of animations as it is of books and movies. In each act, we use tension, anticipation, and foreshadowing to move us through the story, from scene to scene and act to act. In a written story these things may be accomplished by subtle clues, but in a visual story we can accentuate these clues by taking advantage of the medium to create tension with music, with effects, with the style of animation used to appropriate effect.
Using sound to your advantage.
What's more effective: writing bang, or hearing a bang? Animation lets you create an audiovisual experience in which, rather than learning about things happening, the viewer is immersed in the actual event through sound. This can be something as simple as sound effects on a busy street, to the tone of voice used in conversations to build the emotion in a scene and tell us how characters are feeling. Music, too, has a very strong effect. Background music and soundtracks go a long way to setting the tone, especially music that's timed to the motion so that it seems as if the characters are moving in choreographed compositions. Music has a strong effect on people, and can deeply enhance their visual engagement with the story.
Using motion, zoom, angles, and perspective to your advantage.
Visual storytelling provides a unique advantage: use of extremes. Exaggerated motion, dramatic zoom effects, extreme angles, and plunging perspectives can drastically alter the mood conveyed by a scene, heightening the intensity and often creating an almost surreal effect. A high dramatic angle on a scene can make us feel like we're in the perspective of someone watching from a hidden vantage point. Exaggerated motion, such as supernatural speed, can make characters seem deadly and otherworldly. A tightly focused zoom on part of a character's face can heighten the drama of their dialogue. A dizzying perspective can convey the gut-wrenching vertigo of a drop.
Understanding dialogue-driven and action-driven storytelling.
In animation, you don't have the luxury of exposition to convey backstory, character information, plot details, or chain of reasoning. So we have to rely on dialogue and action to do this; much of this involves discovery through conversation, observation, and participation. Conversation will let characters talk to each other to find out about each other, work out problems, or share key plot information. Observation lets our main characters see things relevant to the plot happening in their environment. Participation involves learning information about the story as the characters are experiencing it. All of these take the place of exposition and inner monologue to give us the details we might normally find worked in between the lines on a written page.
Using emotion to convey things without words.
We shouldn't need to be told how someone feels in a visual/auditory storytelling medium. We know someone is in pain by how they clutch the affected area, and their agonized cry; we even know the intensity of the pain by how loud the cry is, and how extreme their reaction. An agonized scream tells us to be worried, while an "ouch!" tells us something is a minor inconvenience. A flinch says papercut; collapsing to the knees says mortal wound. You don't even have to show the wound; it's all about the character actions and using them appropriately to convey emotion. A character can say "I love you" with a look without needing to say the words out loud. Though in animations, you also have the advantage of cartoon effects - fluttering hearts in the background, throbbing hearts bulging from the eyes or chest, etc. Motion, too, has its place; instead of saying "I'm dizzy" a character can reel back and forth with a slack, cross-eyed expression. Take every chance you can to show us how characters feel, whether through normal or extreme portrayals, rather than telling us.
Using color to set the tone and convey the mood of the scene.
One advantage you don't have in verbal storytelling is the use of color palettes to set the tone. Using color in deliberate ways can go a long way to convey the mood; a washed-out palette can convey a drab, gloomy scene, while dark, stark contrasts tell us we're in a hard-hitting noir setting or a horror. Pastels convey a lighter tone, while consistent use of warm tones can tell us we're in a romance. Think about how color affects your mood and what colors you associate with particular emotions, and choose the color palettes and contrasts for your scenes and settings appropriately. Understand, too, how lighting can affect the mood and the setting, whether soft or dramatic. A scene painted with soft gold or red-gold overtones of light can tell us it's sunrise or sunset; blues tell us it's night; deep shadows with the occasional stark edge of a shape can tell us it's night, or in the middle of a thunderstorm, or just...well, really creepy.