Virtues (in the Aristotelian tradition) are qualities that humans alone bring to the world -- and therefore explain, in some sense, why we are here.
Classical virtues include courage, honor, loyalty (to other people), applied intelligence, aesthetic sensibility, and sophrosyne (an ability to find the right balance when virtues conflict).
You act inconsistently with your purpose as a human being if you sacrifice one of the virtues. Virtue ethics says that you should compromise a virtue only when necessary to preserve another virtue.
Integrity is (literally) wholeness. You lose integrity when your actions are not virtuous, because they conflict with who you are as a human being.
For a fuller explanation, watch the video Rational choice II (or read the transcript). It is part of an online ethics tutorial that consists of the following 30-minute sessions: