Jounin are the elite of a village's ninja force. They are the best conventional forces a village can deploy and are death on the battlefield. Their skills are strong enough that they are typically assigned solo missions and as a group are largely a match for any opponent they encounter.
Responsibilities
Jounin receive the most responsibility. They can act as team leaders for Chunin, and some serve as sensei for teams of Genin fresh out of a village's Academy. Jounin are given the most difficult missions, usually only taking A-rank and S-rank missions. It is from the Jounin that village leaders are pulled from, and in times of war others become the captains and generals of armies.
Becoming a Jounin
Becoming a Jounin is one of the most demanding exams a ninja takes. They need to have B-rank or higher skills across the board in the core skills to even be considered for promotion, and then they undergo a grueling series of tests and review of their entire career and personnel jacket. If they make the cut after all of that, they are given a mission - either real or fake - that they'll perform. The mission report by an observer or two on the team is used to further weed out candidates. The candidates still in the running are presented in a report to the village's leader, who then approves or denies promotion of each candidate.
Tests as a Jounin
The testing for promotion covers around half of the overall aptitude battery of tests, and so records are generally updated in the applicable areas for a ninja after an attempt to make Jounin. As they are at the top of the ladder, there are no further mandatory testing or promotions.
Training in specializations is still available, and the final levels of certification for some specializations becomes available. In addition, all new Jounin have to take the Intermediate level of the Ninja Villages specialization.
Typical Stats
Jounin, being the elite, are generally all B-rank and A-rank in their core skills. Veteran Jounin begin moving into having S-rank skills and shifting their B-ranks into A-ranks. These veteran Jounin are some times referred to as "Kage level" ninja.