When the father tells his sons about joining the army, "Obey them, but remember your name and don't take nothing from no man." what does he mean?
THis quote came from The Tall Men by William Faulkner. This part of the story is when the father has just talked about his war experience and how he wants his sons to try to sign up for his calvary, but they have to go sign up for war because they forgot to register and could otherwise get arrested, even though there is no war going on at the time. Their father tells them to obey their sergeants and learn how to be a soilder.
"The Tall Men" is a short story that Faulkner wrote to try to promote the ideas of patriotism and of the importance of rugged individualism. It is in this context that we should understand the quote that you cite.
What Buddy McCallum is telling his sons is that they should not lose sight of the fact that they are individuals who matter. America, as a democracy, is of course built on the idea that all individuals have value. McCallum is telling his sons to remember that and to fight for that idea, even in the face of a government bureaucracy that is sometimes dehumanizing.
By telling his sons to remember their names, he is telling them to take pride in who they are. By telling them to obey, but also to not "take anything" from anyone, he is telling them to do what they are supposed to do. He is telling them to be respectful, but not to simply obey others blindly.
In these ways, McCallum is making a statement about what he believes an American man should be. This is why he gives his sons the advice you cite.