That old staple rule of showing vs. telling. We've all heard it. We've all said it. The difference between relating a dry story, and bringing to life a vibrant, emotional one lies in the clever prose we employ. I love this Maya Angelou quote - all the words in the world have been used before. The way we use them is what sets us apart as authors.
When it comes to emotion, the idea is to bring the audience into the scene. You want them to think "I've been there. I've stood in that character's shoes." And the best way to do this is to approach the writing as if you were directing a physical performance.
Think of it this way: If you're watching a movie, you can read the characters' emotions. They don't say, "I'm angry." "I'm annoyed." "I'm afraid." You hear them yell. You see their hands tremble. You watch their eyes flash.
The same is true of written actions. As the writer, you describe the character's state of being, and the audience can read the emotion in them. Not only that, but because you've described it so well, they can "see" the emotion too. They can feel it.
Properly showing action and emotion makes your story more interesting:
Ex. 1: She was sad when she remembered him.
Ex. 2: His ghost was everywhere: in plate glass windows, walking alongside her; in the faces of every man who smiled at her. When she walked into Starbucks, her eyes went directly to their old favorite table...and glazed over with tears. She left without her ritual cup that morning.
The first sentence gets the job done. It gives the reader the appropriate information. But it's boring. And it doesn't tell you how deep the sadness runs. I can be "sad" because I dropped my doughnut in the parking lot. Or I can be so heartsick I can't bear to keep up with a routine because the coffee shop brings up too many happy memories. Big difference.
This is the fun part of writing. This is where your creativity comes out to play; where you experiment with words and make them your own. You become the trainer dictating their behavior, and thus you find your style.