Character is revealed when a person encounters an obstacle. Think about it: we live our lives, day to day, until that's no longer an option any more. Something prevents us from living life the way we always have. Our routine is interrupted.
I tend to think people's lives are defined by moments. Sometimes only a handful of moments. Michael Jordan makes the buzzer-beating shot. Breana's pregnancy test comes back positive. You name it.
So in a good story we often want to see these moments of no-return, but for comedy we don't want to focus on the stories of enormous magnitude because dramatists already have this market covered. Instead, we want to focus on the day-to-day obstacles that still reveal character because ultimately any obstacle will reveal one's character.
What did Napoleon say when their army ran out of toilet paper? Nobody knows, and the story of their invasion of Russia is lesser for it. What does your partner say when you run out of tooth paste? Were you using too much? Or did they throw it away without doing the roll-and-squeeze method? These comedic moments crop up everyday, even though at the time they may more closely resemble a petty drama.
Writers often confuse the obstacle for the character. The character is the only vehicle by which we can tell a story, certainly in improv (remember our frame of story, and the rule that the only prop you have is your partner?). Sure, a photographer can take a picture of a flower and tell a story, but even then we find we identify with the flower, anthropomorphizing it, translating it into a character if there is to be any 'story.'
When I sit in on writers' rooms, or even when I watch big budget films that were the result of these rooms, I see stories that are flat because they are focused on the obstacles, instead of the characters. You don't need many or big obstacles. You don't need huge stakes of the world ending. You don't need a litany of this happened and then this and then this and then this. These writers become obstacle-machines, and we end up watching a story of obstacle-lists, rather than character reaction, recognition, and transformation.
Any obstacle can reveal character if done right, and as a player it is a matter of listening to see what an obstacle can reveal about our character, be it on the stage or in the modest dramas of our little lives.