The purpose of the "local provider" is to provide a testing ground or sandbox for users to experiment with Juju and to speed up the process of writing charms. Although Juju is intended to work on bare metal (via MAAS) or through a variety of cloud providers or your own private cloud, it can also be configured to run solely on a local machine by means of containers or virtualisation.
Currently, the only stable implementation of this is using Linux containers, LXC (see linuxcontainers.org). However, this option isn't available to those running Juju on Mac and Windows systems so further methods of containerisation and virtualisation are intended to be added. These are collectively referred to as using "the local provider", even though the actual underlying technology may be very different. Whichever method you use, ultimately it should be largely transparent to the user.
The following pages cover the different local providers available: