Jungling is probably the hardest role IMO. You need to have the best map awareness of anyone since you will be deciding what lane to go help out, if you can dive the tower to get the kill, or if a lane is really pushed up. Objective control (dragons and baron) falls to you as well, since you are the one that carries the summoner spell Smite (390-1000dmg to minions/monsters depending on your level) Champions in the jungle usually have a gap closer, or CC. The most popular junglers are
Elise (has been since her launch about a year and a half ago now) amazing in every aspect, great damage without having to build damage (because of the % HP her Q does in both human and spider forms) good jungle camp clear times since she can have two spells at level 1 (take W first), good ganker, E is a stun in human form, in spider form good gap closer.
Lee Sin, huge damage early game, not gated by mana (he uses energy), good ganks (gap closer and a slow)
Evelynn, little different playstyle with her since she is invisible unless enemy is within a certain radius of her or you attack. Mostly involves flanking behind people and surprising them. This makes her my favorite, don't have to worry about their wards, you can just walk through them without a care.
I can go on but I won't