【Beg】 , 【entreat】 , 【beseech】 , 【implore】 , 【supplicate】 , 【adjure】 , 【importune】  mean to ask or request urgently.

【Beg】 suggests earnestness or insistence especially in asking a favor.

【Entreat】 implies an attempt to persuade or to overcome resistance in another especially by ingratiating oneself.

【Beseech】 implies great eagerness and often anxiety or solicitude.

【Implore】 , often used interchangeably with 【beseech】 , at times suggests even greater urgency in the plea or more manifest anguish.

【Supplicate】 adds to 【entreat】  the suggestion of fervent prayer or of a prayerful attitude.

【Adjure】 implies an injunction as well as a plea and is strengthened by the expressed or implied invocation of a sense of responsibility or duty or of something sacred. It may sometimes suggest little more than urgency or peremptoriness.

【Importune】 commonly suggests repeated attempts to break down resistance and often connotes annoying pertinacity.