For anyone interested, this article explains what the "Deep State" is and why we should be aware that it exists:
The Deep State Is A Very Real Thing
This entity is called the civil service, and it was created to limit the power of the president.
There is something ominous-sounding in the deep state. It implies that beneath constitutionally ordained systems and principles, there is a deeper and more potent power in control of the nation. It implies a unified force deeply embedded in the republic that has its own agenda and the means to undermine the decisions of elected presidents and members of Congress. Its power derives from control of the mechanisms of power and being invisible.
The deep state is, in fact, a very real thing. It is, however, neither a secret nor nearly as glamorous as the concept might indicate. It has been in place since 1871 and continues to represent the real mechanism beneath the federal government, controlling and frequently reshaping elected officials’ policies. This entity is called the civil service, and it was created to limit the power of the president.
That's a pretty good article.
The "deep state" is a conspiracist's term to make something normal and reasonable seem sinister. It refers to a real thing. A shallow state would necessarily be a dictatorship. But it's really quite simple: The government is not designed to turn on a dime at each election. The vast majority of the people in the government are neither politicians, nor political appointees, which, if you think about it, is a good thing, if you contemplate the prospect of appealing to a politician to honor your rights and treat you fairly at every turn. And that's how the Constitution and the many laws have established the government set it up, and it's explicitly within the control of the legislative process of Congress to change how the system works. Democracies are built around institutions and consensus, ours included. And this is why the people who work for the government, either in the civil service or the military, swear an oath to the Constitution. Because serving the country is not serving a person, it's serving an institution. And it's also why the fact that the man who purports to be President has on at least a few occasions attempted to get people working for the government to swear an oath to him personally is very concerning.
https://www.law.cornell.edu/uscode/text/5/3331
https://www.nytimes.com/2017/05/11/us/politics/trump-comey-firing.html
So, why are we hearing so much about the "deep state" in the last year and a half? Because the country is under attack by people who find its institutional depth to be an obstacle. On the one hand, you have Trump, who is determined to have one-man rule under himself, and who is contemptuous of courts, members of Congress, officials in his own administration who are meant to be independent, anyone who might oppose him. He's set up a state-within-a-state of lackeys he can place in official roles, and family members. And on the other hand, you have Russia, which is running an intelligence op against us, deploying a massive amount of agitprop, and buying or building propaganda networks in order to convince Americans to tear down their own government and to undermine their trust in independent sources of information.
http://www.newsweek.com/sinclair-broadcast-group-must-run-deep-state-rt-russia-today-867029
https://www.politico.com/magazine/story/2017/03/steve-bannon-deep-state-214935
Don't fear the "deep state". The threat is from the people who want to rule over you without democratic institutions or processes to protect you.