For forms in France. And also letters you receive from official sources like a French government agency.
NOM is nom de famille, or last name or surname.
Prénom is first name.
And just so you know:
It is useful to know that in French official-type documents, they always put the last name in all caps and first. So you get: SMITH, John in lists.
That's why nom is written NOM and Prénom is in lower case.
If you have a middle name that you commonly use, the only place to put it is with your first name like this: Jean Marie. Or you can just leave it out if you don't have one you commonly use. It's true that then it will be interpreted as a first name with two parts in France. Them's the breaks. :)
[Here's a little story: I used to work for the French-American Chamber of Commerce in the States (not saying which city). The president sent out official letters to people like the CEO of Chase Manhattan Bank and he insisted on using all caps for the last name, like this: Dear Mr. SMITH, I tried to tell him many times that it was simply not done in English, and he did not believe me. So embarrassing. :)]
PS: This is for France. I am unfamiliar with how they do things in other French-speaking countries.
NOM et Prénom: SMITH John [not sure of how exactly to write that, I would have to see the actual form.]
There is a famous French movie by Louis Malle called Lacombe Lucien where the name of the character is actually Lucien Lacombe but the filmmaker reversed the name on purpose to imitate bureaucracy, and maybe more... It is about World War II [I will leave it up to you to read about it.]