The deep state is a hypothetical group of people and/or institutions that have undue, unofficial, and publicly unacknowledged power over government policy. It is most often said to be composed of members of government agencies, particularly including the military, but theories vary. It is not uncommon to include industry leaders and influential members of the financial system.
The term originally referred to shadow governments in Russia and Turkey, but now it is commonly used in discussing American politics as well. This is bordering on conspiracy theory nuttiness, except for the fact that a considerable amount of political power is wielded behind the scenes, and the phrase deep state is often not more than a cynical way of referring to lobbyists and the entrenched old boy network.
In many cases, you might prefer to use the term state within a state, which refers to governmental institutions that do not answer to to the civilian political leadership.