The Imperial Galactic Government was the reorganized government of the Milky Way galaxy following the economic collapse of the Galactic Empire. The term Imperial is now an anachronism, however, as the hereditary Emperor is nearly dead and has been for many centuries. In the last moments of his dying coma, he was locked in a stasis field which keeps him in a state of perpetual unchangingness.
All his heirs are now long dead, and this means that without any drastic political upheaval, power has simply and effectively moved a rung or two down the ladder, and is now seen to be vested in a body that used to act simply as advisers to the Emperor - an elected government assembly headed by a President elected by that assembly. The President in particular is very much a figurehead - his real purpose is not to wield power, but to draw attention away from it.
Bureaucratic agencies under the Imperial Galactic Government include the Interstellar Revenue Decimation Service, in charge of taxation, and the Galactic Hyperspace Planning Council, in charge of hyperspace routes. Vogons are employed to fill most of these positions, mainly because no one else is willing to do it and the Vogons want something to complain about. The Government protects its assets with a space navy known simply as the Galactic Fleet.
As the Imperial Galactic Government became ever more mired in bureaucracy, its power was threatened by the emergence of the Federation League of Planets, which sought to take control. After the IRDS agreed to work for both organizations, the Federation had enough resources not only to do so but to fight a war if it didn't get its way. The Imperial Galactic Government was never formally disbanded but became little more than a figurehead.