Sorry if this is a really broad question, but what exactly does a "national emergency" technically do in the United States of America? How are each of the branches of government affected by this state, and what is the impact on normal citizens?
The short answer: It depends.
It's not like declaring martial law. There's no generic "national emergency" under the National Emergencies Act and in the past national emergencies have been declared for "Prohibiting the Importation of Rough Diamonds from Sierra Leone" or "Continuing Certain Restrictions With Respect to North Korea and North Korean Nationals." The Brennan Center lists 58 previous emergencies.
The president must specify the grounds for the emergency and the statutory provisions under which the president or his agents will act.
For example, in 2014 President Obama declared a national emergency "Blocking Property of Certain Persons Contributing to the Situation in Ukraine," which imposed sanctions "on specified persons operating in sectors of the Russian economy identified by the Secretary of the Treasury" in response to the Russian intervention in Ukraine.
In the document, Obama specified the findings justifying the emergency, cited the laws that allowed him to declare the emergency, and spelled out how the Secretary of the Treasury would work out the detailed implementation, noting "All agencies of the United States Government are hereby directed to take all appropriate measures within their authority to carry out the provisions of this order." Individual Americans were affected to the extent that they could not do any business with the specified persons without facing legal action themselves.