Deep State or The Shadow Gorvernment is the term used to describe a group of government officials who wield disproportionate power over the government despite not formally being elected/in charge.
An example of the Deep State exercising its powers was shown during the Podesta email leak, where in 2008 Citibank executive Michael Froman sent Podesta a list of preferred cabinet appointments a month before the election occurred. Obama’s eventual cabinet exactly matched Citibank’s suggestions.
More recently, the term has been used in relation to Trump. Independent media has been reporting that Trump and the Deep State are butting heads, and there is speculation that Deep State is positioning to impeach Trump.