On defense in American football, a pass rush is charging across the line of scrimmage towards the quarterback in an effort to stop or "sack" them. The purpose is tackling, hurrying or flushing the quarterback out of their protective pocket or the play's design.
"Pressures" and "hurries" are terms used to describe pass rushes which successfully alter a quarterback's performance on a play - forcing them to throw before they find their best target or scramble out of the pocket to elude being tackled - that fall short of completed sacks.
A related form of defensive rush is to disrupt or sack a kicker attempting to kick a field goal, extra point, or punt the ball. On offense, "rushing" is to run forward with the ball to gain yardage.