Question: Is it martial law or marshal law? Why?
Presumably you're referring to the system of rules that takes effect when the military of a country takes control of the normal administration of justice.
It's martial law.
The term martial is an adjective and basically means "of or associated with the armed forces" or "warlike." The word marshal, on the other hand, is a noun and refers to an officer of the law who performs duties similar to those of a sheriff.
The inhabitants of P93-678 were a martial people.
Describing Deputy U.S. Marshal Mary Shannon's family as dysfunctional is a major understatement.
The two terms are pronounced exactly the same way, which is no doubt why they're often confused. To help you remember which one is which, think about how each came to be a word, or its etymology.
For instance, the word martial is based on the word/name Mars, the god of war in Roman mythology. Thus, thinking of an association along the lines of the following could be helpful to you: mart- --> Mars --> war --> warlike.
While there are references to marshal law out there, it is an error in usage, especially in article titles, or an intentional play a person's name or position. For example: this comic book and this movie.
The correct usage is martial law.
The Oxford Dictionary of Word Histories (Glynnis Chantrell)