What a Junger's role SHOULD be is to fill out a need for your team. Bot/mid lane isn't the best for peeling away from lane to secure objectives like dragon early? Grab someone who can do so. Your team needs a hard frontline, grab one of the tanks that can do so. Top/support is going tank, but you still need a bruiser? Grab one of those. Your frontline is good, but you need an assassin/carry? Those can work too. You have lanes that can snowball easy, they just need a kill or two ahead to get rolling? Grab someone who can fill out what they need (as in damage or cc, probably the cc). So in this sense, they are a lot like supports, in that they should be working with their team to shore up it weaknesses, but they have more flexibility in the sense that they can fill in more weaknesses, because they do have a steady source of xp/gold income, so they can play item based carries.
One of the major things that junglers need more than anything is speed. If it's just high mobility, fast clear times, or great dueling potential for those who counterjungle (IE: fast champion kill times). These are the marks of a good champion for jungling, and knowing your champions strengths and weaknesses is key. There are champions who don't have any of these that are good junglers, but they bring something else to the table that is stronger than their lack of speed, usually in the form of being healthier than others once they do a full or partial clear of their jungle.
Because of the speed, and their "lane" wanders across almost half the map, they were expected to, the same way that a support is, to put up vision, and to recognize and exploit mistakes from the opponents. This really gets me going for both the role of support and the role of jungle, because apparently you are supposed to have pro level of threat/weakness recognition, and the split second reactions/mechanical execution to respond to these situations. When I'm playing with people at my level, I'll make the same mistakes, same misclicks, same poor shot calls as them, but because of my play immediately affecting them instead of say, 5-10 minutes from now when my lane opponent decides to roam after they were just a better player and won the lane, they react instantly.