It is said in the wiki that:
...there was mention of a Jōnin Exam in the anime's Kurama Clan Arc.
Does anyone here know what are the stages/steps/missions in order to be a Jounin?
I think recommendation to become Jōnin ninja are send by villagers and the hokage has the final decision of a making a one.
I'm not sure about this but I think that the candidate has to create a new jutsu in order to become a Jōnin (like Kakashi with Chidori, Minato with Rasengan).
We don't get any other details other than this. Same for the exam. I think that the created jutsu will be examined.
My answer is based on the anime.
From the wiki
Jonin are generally highly-experienced shinobi with great individual skill who serve as military captains. They are often sent on A-rank missions, and experienced jōnin may even be sent on S-rank missions (which are considered to be the greatest difficulty). It is not unusual for jōnin to go on missions alone. Jōnin are generally able to use at least two types of elemental chakra, proficient genjutsu, and decent taijutsu skills.
We can deduce that to become Jounin a ninja needs to meet certain requirement such as:
1) Must be highly experienced and skillful
3) Capable of handling at least two types of elemental chakra, proficient genjutsu, and decent taijutsu skills
Once this criteria is met:
The villagers may send their recommendations for a jōnin appointment. The Kage reads said recommendations and may consider appointing an individual a jōnin if his or her qualities are recognized as adequate by enough people and himself.
About the exam, I'm not sure since I don't follow well the anime. About creating a new jutsu, I have found out that each technique is well classified according to their rank. The link shows list of A-rank Techniques. This may support a theory that a Jonin is required to have at least a few A-rank jutsu's and may be S-rank Jutsu's.
From an answer to a similar question :
In the Road to Ninja movie, Naruto tried to get a recommendation from Iruka for promotion to Jōnin, but was refused. Iruka said that Naruto had to become a Chūnin before he could become a Jōnin.
Iruka: so, what did you want to talk about?
Naruto: Um, Iruka-sensei, uh, listen, you think you can submit a Jonin application for me?
Naruto: It's just that all of my other classmates are going to have their parents fill out applications for them, and, well, I, uh...
Iruka: Nope, sorry. I won't.
Naruto: But why not?
Iruka: Because that step is supposed to come after you become a chunin. Sure you saved us from Pain, and everyone calls you the "Hero of the Hidden Leaf", but I can't give you special treatment. I mean, even your Dad worked his way up through the ranks - from Genin, to Chunin, to Jonin, to become Hokage. Then he went on to became a Hero.
Naruto: hmm, to become just a face carved into a rock.
Credits to Remy Lebeau.
So we know that being a Chunin is essential , combined with the other factors from the other answers given here.