In response to Putnam:
There are a couple different breakdowns, between farm junglers vs. gank junglers (the latter have been in-meta for a while), between junglers with a high-impact early game (Lee Sin, Pantheon, etc.) who gank early and often to snowball their laners vs. junglers who scale into lategame, and so forth.
I'm by no means an expert, but I'll do my best, specifically in regards to what's the popular meta right now. The general idea behind most of the popular junglers is that they can come out of the jungle as early as level 3 (the first two buffs and a minor camp), and that they're supposed to apply pressure, either to snowball a lane or to ease pressure from a lane that's tilting against their team. The exact mechanics vary from champion to champion, but there are various goals: killing enemy laners, forcing enemy laners to back, burning their summoner spells, counterganking, and covering lanes when people are forced to back at inopportune times are some of the most important.
A good jungler has to consider the consequences of their actions; is it more important that they get a kill or that their laner does; is it worth risking a death to tower-dive an enemy laner; is the risk of counter-jungling worth the benefit of denying the enemy jungler their farm/stealing buffs, &c. There's also something of an unstated element: putting one player in the jungle not only brings the benefits of ganking, but also means that both top and mid lanes get their full lane exp and cs, which is more valuable than the increased pressure in a single lane. There's also a fairly substantial degree of mindgaming inherent with the dynamic of both teams having a player whose position is typically unknown; is my enemy taking an aggressive posture in lane because they've got a gank coming, are they backing off to try to pull me farther towards their tower for a gank, how can I trick the enemy into a vulnerable position for my jungler, do they have that brush warded, is that Thresh lantern in the brush a ruse or just a really obvious gank, &c.?
Later in the game the jungler shifts roles; they start acting as whatever they've built as, typically either an assassin like Kha'Zix and Yi, or something on the fighter-tank gamut like Jarvan IV and Vi. They've also got an important role in terms of objectives; the summoner spell Smite is the most reliable way of securing kills on the Dragon and Baron, both defensively for a team looking to finish without losing it, and for a gutsy jungler trying to steal it.
But yes, the common factors between pretty much all viable jungle-worthy champions are as Stuebi said: Sustain, clearspeed, and gank potential. The last is probably the most important in the current meta, and also translates well into lategame. A lot of the current in-meta junglers have a similar feature in that the vast majority have high-impact CC/displacement and/or a gapcloser combined with good burst damage.