The things i find are most important for jungling are:
-good sustain in jungle to be able to gank at any time
-fast clearer in jungle to get buffs early
and most importantly the ganking ability:
Junglers must be able to have a way to have his ally gang up with him, which means slows, stuns, fears, snares ect...
Some good examples of junglers are:
Udyr: He has bear stance which gives him a boost of speed and stuns the enemy when hit, perhaps one of the fastest junglers depending on what stance you go with, but he is tricky to play well with.
Mundo: His Cleaver slows and does good damage, I find since he is tanky he can do well if he ganks a 2v2 bot and can take the damage until his allies come up. He also clears very quickly with his e and w.
Warwick: ugh he is a tricky ganker b/c before he is level 6 he has no slows, snares, or anything besides his e, which makes him faster but that depends on the enemy being a low health. Also some junglers who go warwick go exhaust sided with smite to slow enemy, but that sacks the ability to get an escape. However warwicks lifesteal is so good when jungling he clears very quickly.
Olaf: My favorite jungler because of his massive burst of damage. Olaf has a full set of abilities that just make him great at jungling and ganking. Olaf depends on his q to land to slow the enemy, however if olaf goes over his axe thrown he has it pretty much has his ability reset to throw again. Along with that he has his w, which increases his damage and his e which just destorys target health with the true damage. Once level 6 he ignores all cc on him which allows him to plow through attempts to slow or stun olaf. I usually side ghost up with olaf while jungling and laning because with his ult and q he will quickly catch up and destroy his opponent.
Riven: A great ganker if the stuns are landed correctly with the e to get close sided with her q she can get a quick stun and then w the enemy to finish the job. By then you will probably have an ally to help slow the enemey champion because most champions have some sort of cc effect
Shen: He is able to lunge far and have a great taunt which is just great for ganks i mean if the enemy is just standing still what could be better?
Rammus: imagine now a tank who is going a amazing speed briefing stuns when he hits his target then taunts while becoming a invincible rock. Rammus is banned so much because of this, but if you get a chance take it!
Master Yi: A good jungler overall with his abilites and his ulti later on, however the thing about master Yi is that if he is fed while ganking in jungle he snowballs and dominates the game. Yi has no slow or such, but with his ult he can make up for the lack of cc and sided with his q
Amumu: A good gankers sided with his wrap which briefly stuns and allows him to pull himself to the enemy if he lands it correctly. When he gets his ult that is where he shines as a ganker.
Fiddlesticks: one of the most interesting champs because he seems to not be played often and yet has a full set of abilites to just wreck havoc when ganking... His q fears, his w slows and regains health which allows him to clear quickly and remain at top health (must have blue buff however :/) and his e silences the enemy. Sided with his ult he just dishes out massive damage.
There are other junglers, but these were the ones that came to my head atm