When the left used the term, it usually implied the intelligence gathering apparatus of he government that was used to disrupt civil rights movements like how J. Edgar Hoover used the FBI to gather dirt on MLK and Malcolm X or how COINTELPRO was used to disrupt the Black Panthers.
Normally, as others have said, it is the bureaucracy that functions between administrations and Congressional sessions. The career people who work at State or NSA, or CIA, or FBI, or DOT, etc... who work under Republicans and Democratic governments alike and keep the lights on and the checks going out.
On the fringe right the Deep State was the "black helicopter" people who were, "coming to confiscate guns" or who were going to do infringe on the "Posse Comitatus Act" and create Amero merging the US, Canada, and Mexico.
More recently, the right wing, having merged its mainstream and fringe elements, have taken to calling anything that doesn't overtly support the right wing agenda as Deep State. The FBI and CIA, which were sacrosanct during the Bush years are now the enemy because they are looking into Trump's crimes.
It is an ambiguous term that can be used as code for anyone who wants to project their distrust of government into.
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