By definition a brush is a device that has bristles, wire, or other filament that is used for cleaning, grooming, or painting. In the context of the internet, it generally refers to the digital version of the paint brush, for use with a digital imaging program, most often with Adobe Photoshop, or Corel Painter.
While traditional artist brushes are varied, the variance of the digital artist brush is electronically handled within the software. One can use a mouse to perform their painting, but most professionals use a tablet device such as the Wacom Intuos or Cintiq that comes with a pen device which functions like a brush and gives you the feeling of working with a traditional brush.
Because of the popularity of Photoshop and Corel Painter, many sites have sprung up that support brush technology. Some of these give away free brushes, and others offer professionally designed brushes for sale. In either case, the technology of electronic brushes will continue to develop over time.