You get this title by joining one of the four (Bishop/ Mordino/ Salvatore/Wright) mafia families in New Reno and having more than 4 Intelligence. For each mafia family there is a chain of quests, upon the completion of which you become a Made Man of that family.
You can see the special stock and get discounts at the New Reno Arms.
You can paralyze Jagged Jimmy J when you have Intelligence less than 4 and when you want by choosing the line "How can one TRULY know another, Jimmy? Is it even possible while we are on this mortal coil? Or is it only possible when we free ourselves from our physical bodies?"
You will lose the Made Man title if you turn against your own family.
For each family boss, there's a way to assassinate them without getting blamed for it, and killing a boss "silently" or in these special ways will not remove the Made Man title.
It's possible to become a member of each mafia family, just don't become a Made Man of the Salvatores until you've got the quest from Mordino to kill Salvatore.
If you have a family quest active when you turn that family hostile, the quest will be set to failed in your quest list.
Although the in-game description says "you look great in a fedora", no wearable fedoras are present in the game, and they are only seen worn by casino bouncers.
Behind the scenes
Strangely, the sentence for paralysing Jagged Jimmy J when you have IN < 4 it's not a normal sentence (normally poorly constructed and difficult to understand) said by a low-intelligence character and looks like more something that an intelligent character could say.