A plea is what you make when you're begging for something with a sense of urgency and emotion. While you wouldn't call asking for a hall pass a plea, you could make a plea for justice or world peace.
Plea also has a legal meaning. When you get a traffic ticket or if you're accused of a crime, you have to enter a plea of "guilty" or "not guilty." A "plea bargain" is when you make a deal with the prosecutor — you may plead guilty to a charge (by entering a guilty plea) that has less of a penalty and, in return, the prosecutor drops the more serious charge.