Most authors try to understand what a character is feeling at a particular moment: He's angry here; he's happy there. Many authors also consider how the character and his emotions change over the entire story: He begins insecure; he ends confident. But few think about how the character's emotions develop over the course of a single scene.
In my research last spring, I came across a fascinating guide called Book on Acting. The title arises from the author's name, Stephen Book. He's a famous acting teacher. Book directs actors to consider how their characters' emotions develop over a scene. He calls this the emotion arc. I quickly realized this was great advice for writers, too.
A character whose emotions don't develop or change in a scene is static and not terribly interesting. On the other hand, a character who is jumping from one emotion to another in each paragraph is unlikely to seem believable. So it's important to create limited, focused changes.
CHANGES IN INTENSITY
The simplest type of emotion arc is a change in intensity. The basic emotion remains the same, but the intensity of it changes over the scene. For example, a character may feel happy in a scene, but the intensity of his happiness may change, starting, at its lowest intensity, as calm and then building through content, pleased, amused, glad, happy, cheerful, giddy, jubilant, elated, joyous. Generally, it's more powerful to show the emotional intensity increasing rather than decreasing. You usually don't want to start the scene with the emotion too high, because then there's no room to build. When you do increase the intensity of the emotion, it usually shouldn't happen too quickly--unless something startling drives it--so it seems believable. One common weakness in the emotion arc is plateauing, reaching a certain level of intensity and then just staying there. Every arc doesn't need to start at the lowest level of intensity and go all the way to the highest, such as calm to joyous. You could just as well go from amused to happy. But reaching "happy" halfway through the scene and then leaving the character stuck there makes the events in the scene seem less important, since they have no emotional impact on the character.
A more complex type of emotion arc involves both changes in intensity and changes between emotion families. Changes between emotion families Book calls "emotion switches." For example, a character might be on the happiness emotion arc. Perhaps his boss is telling him that he performed well over the past year, and as she makes several compliments, the character's happiness increases from content, to pleased, to glad, to cheerful, to elated when his boss says he's getting a big promotion. The boss then suggests they should go out to dinner to celebrate. This triggers an emotion switch to a new arc, which we might call fear. He might begin at a very low level of fear, feeling cautious (is this sexual harrassment or is he mireading her?), then build to nervous, anxious, apprehensive. Similar to using a single emotion arc, when you have an emotion switch you generally want to start the second emotion at a somewhat low level of intensity, so there's room to increase it. For a character to switch from one emotion family to another, the scene generally has to show some important event to trigger the shift.
You can see a great example of an emotion switch in this scene from the movie Goodfellas. For those unfamiliar with the movie, these characters are mobsters. Tommy (Joe Pesci, the guy doing most of the talking) is in charge, and he has an explosive temper, as has been established earlier. Watch how Henry (played by Ray Liotta, the guy with the cigarette on the right) becomes more and more happy, until Tommy asks, "What do you mean I'm funny?" There you see Henry's emotion switch to fear, and you can see it grow and grow. This is one of the strongest scenes in the movie because of the powerful emotions conveyed and the threat underlying everything.
One more important concept to keep in mind is that of the umbrella arc. An umbrella arc is "an accumulation of separate feelings from different emotion families that adds up to a singular emotional response." Umbrella arcs describe more complex emotional states your character may be experiencing; he may be feeling several emotions at once within an overarching umbrella. These umbrella arcs may be labeled with conceptual terms, such as abandonment, betrayal, or denial, which are not emotions in themselves but can carry emotions. For example, abandonment may bring with it emotions of hurt, loss, fear, sorrow, and anger. The character may be feeling a combination of these at once. He shouldn't be jumping back and forth between these emotions, though. That would feel too random and disjointed. Perhaps he's feeling hurt, loss, and fear, but the fear grows stronger over the scene, or he shifts from hurt being predominant to fear being predominant, due to some turning point.
It's important to acknowledge that people experience emotions differently, so "happiness" may not go through the same gradations for one person as for another. Similarly, people feeling an umbrella concept like "abandonment" won't have the same exact combination of emotions. So you want to discover the emotions characteristic for each particular character.
What I see often in the work of developing writers are characters whose emotions wax and wane several times in a scene, leaving us with no clear sense of progression; or characters whose emotions move from one feeling to the next to the next with no sense of focus or causal connection; or characters who seem to feel the same emotion at the same intensity through the whole scene--or who feel nothing at all. Becoming more aware of how you want to shape your character's emotion arc in each scene can lead to much more powerful scenes.