In general, Beggars Night represents the "treat" portion of Trick or Treat, where children in costume make evening rounds of homes (and to a lesser extent to businesses during the day) and are given candy. This being closely tied to, but distinct from Halloween, when various forms of mischief may occur.
The night has usually been scheduled by municipal governments on a date prior to the actual Halloween date of October 31. In recent years however, the trend has been towards having it on Halloween, and the use of the term has been slightly diminished because of the loss of the distinction from the date of Halloween.
Mid-Ohio Regional Planning Commission's 2004 survey of municipalities on the date of Beggars' Night: http://www.morpc.org/web/eSource/2004-November/BeggarsNightFeedback.htm
Newspaper article on Beggars Night in Seabrook, Massachusetts: http://www.newburyportnews.com/punews/local_story_300124616