The goal when writing fiction, as I and nearly every creative writing instructor keep harping on, is to show the reader, but not tell them what’s going on in a scene. I often talk about how it's best to allow characters to reveal themselves through what they say and what they do, in scenes, rather than telling the reader about them, in exposition. That's often tricky when it comes to expressing a character’s emotion. Here, apprentice writers tend to rely too heavily on telling or interior monologue. The character might come right out and say (or think), "I felt sad."
 
Instead, of course, writers hope to allow the reader to experience that emotion for themselves, so we don’t have to explain what the protagonist is feeling. Again, the old show, don’t tell. But how to do that? The answer to this problem is a relatively simple concept that has a fancy handle: the objective correlative.
 
With the objectice correlative, we express the emotion of a character through the situation or setting around them, offering description of weather, landscape or objects in a room, say, and the character interacting with that environment, in such a way as to evoke emotion in both the character and reader. A cheap and simplified example comes from the movies: when a character is depressed, sad or at their lowest point, rain will run down the window behind them. The world is overcast and gray, expressing the character's emotions. 
But this moment in the rain comes after a series of events have occurred to make that character feel sad. Say my character is beamed up into a spaceship, whisked off through outer space to another planet. We then see her standing in the rain on a foreign planet, looking towards the stars, longing for earth. Without being told what she feels, we sense her deep homesickness.