Integrity is a personal quality of fairness that we all aspire to — unless you're a dishonest, immoral scoundrel, of course.
Having integrity means doing the right thing in a reliable way. It's a personality trait that we admire, since it means a person has a moral compass that doesn't waver. It literally means having "wholeness" of character, just as an integer is a "whole number" with no fractions. Physical objects can display integrity, too — if you're going over a rickety old bridge that sways in the wind, you might question its structural integrity.