I guess I'm asking more or less of the legal definition of martial law and any details surrounding that term?
In the US, probably the best way to define "martial law" is when the privilege of the Writ of Habeas Corpus is suspended, because Article 1, Section 9 of The Constitution states "The privilege of the Writ of Habeas Corpus shall not be suspended, unless when in Cases of Rebellion or Invasion the public Safety may require it". The suspension of law and rights under martial law is not absolute. The Posse Comitatus Act 18 USC 1385 prohibits using the US Army to enforce domestic policies (this does not excluding using the national guard for such purposes) -- if law and rights were completely suspended, this act would not have any meaning.
In the case of Ex parte Milligan 71 U.S. 2 (1866) which is about Lincoln's suspension of habeas corpus during the Civil War, the Supreme Court said
If, in foreign invasion or civil war, the courts are actually closed, and it is impossible to administer criminal justice according to law, then, on the theatre of active military operations, where war really prevails, there is a necessity to furnish a substitute for the civil authority, thus overthrown, to preserve the safety of the army and society; and as no power is left but the military, it is allowed to govern by martial rule until the laws can have their free course. As necessity creates the rule, so it limits its duration; for, if this government is continued after the courts are reinstated, it is a gross usurpation of power. Martial rule can never exist where the courts are open, and in proper and unobstructed exercise of their jurisdiction. It is also confined to the locality of actual war.
The point is that martial law does not equal a complete military dictatorship.
Martial law can be declared by a governor, and we can look to the statutes to determine is there are any limits on such a declaration. In Washington, RCW 38.08.030 allows imposition of martial law, which in the strongest case "is the subordination of all civil authority to the military" (which again is a far cry from "the suspension of all law and rights"). It is hard to know what the limits would be, but it is unlikely that a complete suspension of the Due Process clause or the First Amendment would be tolerated.