To entreat is to ask for something that is really important, like when you entreat the jury to spare your life.
The verb entreat implies that the person doing the entreating is really serious about what has to happen. Maybe it's even a matter of life and death, like when parents entreat their children to never drive drunk. Synonyms for entreat are plead, beg, beseech, and implore — all of these are words not to be used lightly. They carry more weight than ask and request.