A "local environment" is a development environment that exists on a developer's personal computer.
Where Drupal is concerned, a local environment includes a "solution stack". A solution stack is an ordered group of software that makes it possible to complete certain tasks on a computer.
A Drupal solution stack consists of an Operating System (Linux, Mac, or Windows), a Web Server (typically Apache but IIS can work as well), a Database Server (typically MySQL but can work with MSSQL), and the PHP scripting language.
There are several different options and ways that a developer can put together their own solution stack. Web Services suggest, however, that you install and use Acquia's Dev Desktop 2, which installs very quickly and easily and also integrates with Acquia.