Exigence refers to desperate or urgent situations. You might hate to miss a day of school, but the exigence of your stomach ache means you're going to have to forget that perfect attendance award.
To understand the pressing importance of exigence, look to its Latin source, exigentia, "urgency," and its root, exigere, "to drive or demand." Exigence demands immediate action or urgent attention, on a small or large scale. In rhetoric, this word is used to describe the situation or problem that compels someone to make an argument.