Short version
I favor 
to carry games, by virtue of their versatility, reduced team dependency and superb scaling. They can also farm well, which grants much-needed reliability.
(Long version below, find your favorites. I explained my logic as best I could.)
Bruisers
They scale well and can deal damage in addition to being able to initiate effectively. They fill a tank-ish role while maintaining effectiveness through good scaling and have significantly less team dependency than full tanks.
Gragas is one of the best but be very cautious as he is more difficult than you'd think to use at maximum effectiveness. udyr can farm and scale well, and Xin Zhao can build damage, tanky or bruiser. He can also peel exceptionally well, should you get a carry fed.
Tanks
Tanks are tricky, because you can set plays up perfectly, but you still need your allies' help to pull the trigger and reap the fruits of your labor. That being said, I can never say a word against the effectiveness of a strong initiate, and Amumu, Malphite and Nautilus bring it.
Another weakness to be mindful of, especially with late game scaling tanks such as Cho'Gath, Amumu and Malphite, is that they're weak pre-6 and their ganks are sub-par until then. Your allies likely won't help if you're counterjungled, your dueling is fairly poor and your farming is average at best. If your lanes start to fall apart, they can't put out the fires as well as you might hope.
That all being said, they can set up plays and win games. Just be aware, you will need decent cooperation from teammates to reach maximum effectiveness. Cho'Gath is an oddball here. He maintains high levels of damage, some of the best objective control of any tank (You won't be out-smiting his 1k true dmg ult) and he scales exceptionally well.
Mages
Slim pickings right now, but these both scale well, and Diana farms pretty well. Both need 6 to gank well, and Fiddlesticks is a poor duelist. I also question the wisdom of picking Diana, since she's a dive assassin, and certain other picks do this better. Still, she sticks hard and farms quickly. Fiddlesticks in a teamfight is utterly devastating when you land a good ultimate, so definitely don't overlook that. Evelynn's ganking is insane, so if you want to hunt down opportunities, she's solid, especially in lower ELO where nobody knows how to react to her properly.
MADCs
MADCs, or melee AD carries, are probably not very welcomed, but they can carry hard. Yi is hard to master, and the other two are risky. I'd warn away from these, but if you have a tanky top and tanky support, and provided you know how to use them effectively (Practice), they can carry games pretty hard.
Assassins
These are my go-tos if I feel the need to carry a game. They snowball incredibly hard, and they are excellent at picking off lone targets to get a lead. They can build bruiser (Which I strongly recommend after you get some damage items to prevent dying and giving the lead back) but these guys are always a threat. Nocturne and Rengar farm well, can kill a fed enemy very well, snowball hard, scale well, teamfight well and are the two best pick assassins in the jungle, in my opinion. Land some picks, take objectives and you can snowball unfathomably hard.
Shaco is the odd man out here. He's a control jungler, but at the same time functions like a carry jungler AND a pressure jungler. He kind of fits into all three categories of the jungle trinity. He's also incredibly hard to learn, insanely hard to master (See what I did there?) and very much hated by both teams. However, there's no denying a competent Shaco will piss off the enemy team and carry a game. His split pushing is fantastic and his ganks are some of the most terrifying (I'll be here all week) around.
Final Thoughts
Really control the objectives, which is dragon early on. Ganking bot/mid the most is ideal, as it will allow you double up on rewards. Get a kill PLUS dragon. Top is fine, if you see an opportunity, and the top lane snowballs pretty often, but your focus early game should be to get towers, dragon and kills.
Also, NEVER FORGET TO CS. They are worth several kills worth of gold. You should aim for at least 80 CS by 20 minutes. I usually aim for 100, but it will depend on the game. If you're ganking and securing objectives frequently, your farm will be lower. Remember, if you aren't in the middle of ganking a lane, you should be CSing.
I prefer bruisers and assassins, since bruisers can go bruiser or full tank, and assassins can focus on damage or go bruiser. That versatility is really nice in my opinion. That said, some swear by tanks and mages, so just find what you're comfortable with.
Finally, wards are control. If you ward the jungle and objectives, you can control them far more effectively. Never forget to upgrade your trinket, and wards will both reduce deaths and increase kills. Don't go overboard, but they really, really do help.
If you have any questions, feel free to ask. I have over 1k games as a jungler.
