The Imperial Republic is an online community whose main purpose is to encourage the roleplaying among its members, providing infinite opportunities to have fun and entertainment.
However, most of the online communities take place in a timeline that is basicly focused around the Star War movies and the universe they portraited. However, the Imperial Republic went further on that concept. Our timeline takes place several years after the Galactic Empire era.
As you can imagine, the Galactic Empire met his end at the hands of the rebellion, and later, the New Republic was founded, replacing the previous government. However, the vices that led to the end of the Galactic Empire somehow found their way to the new government and when added to new vice, finally they made the new government collapse.
Several years passed where chaos was the only ruler in the Galaxy. Slowly, a group of dedicated people led by James Stratus started a reconstruction effort that led to the foundation of the Imperial Republic.
You can have more information in the Historical Records of the IR, but this should have given you a general idea of what is the universe you have entered. The official timeline is located here.
As in terms of organization, the Imperial Republic is a mix of and Imperial and a Republican system, since is ruled by a Throne, but also has a Senate. The other institutions are mostly the same as you would find in any Imperial like government.