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.
 
Here 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.
 
Obviously, this example of the sad character standing in the rain is so overused that you want to avoid it. But that's objective correlative in a nutshell: not just one object or event (like the rain) that evokes emotion, but an accumulation of elements that together evoke emotion. Rather than say, "she was homesick," the reader is simply shown the events that lead to her sadness on the planet, and then in the moment of the rain falling as she looks to her home world, we understand she is sad and perhaps feel it for ourselves.
 
Having said that, just one object in a scene can express the character's emotion or state of mind. That's also the objective correlative in action. For example, in my novel The Spawning Grounds, a nature or water spirit surfaces into the human world, taking possession of a boy. Here's the moment where that spirit "arrives:"
 
"The boy followed a blue wall with his hands and came to a pool of water hanging on it. He attempted to slide his hand into the water but his fingertips wouldn't penetrate. In the hard surface of the frozen water, he saw the face of a stranger."
 
I could have simply said, this water spirit was "confused" by the human world. Instead, I wrote about his encounter with a mirror which he doesn't understand. We sense his confusion as he quite literally faces his new self, the boy he's taken possession of. On another level, the moment expresses the pain and confusion of mental illness that the human boy also feels.
 
In short, the objective correlative is all about using situation and setting, and the objects found there, to express character emotion, so the reader can sense what the character is feeling and perhaps feel it for themselves. That way, we don't have to tell the reader what our character is feeling. It's the old show, don't tell.

For more, check out the link Show, Don't Tell: Applying T.S. Eliot's Objective Correlative to Fiction and Poetry.