Jungling is about two things: buff control and objective control. Some people consider them one thing ( buff control = subset of objectives ) but in this case I'll explain them in two different things.
Think of yourself as a secondary support/substitute laner. When your laner scores a kill and is about to go back you go into their lane and shove it so your team's minions die under the enemy tower which is denying the enemy laner gold and experience. If your laner dies you can go to their lane and get the kills on them so the gold and experience is not wasted. If an enemy is over-extended in a lane while your allies are still in decent health become that second or third laner and attack the enemy from behind. If you find you don't need a buff at some point ( Nunu does not need red buff after his first or second clear ) try giving it to your ADC or top laner and a similar action can be taken with your blue buff. If one of your laner starts seizing an objective go join them; help them.
The reason you are in the jungler as a jungler is actually two reasons: buff control is the first reason while letting your laner get better exp/gold is the other. Having a second duo lane will deny your team a lot of gold and put them behind so sending somebody into the jungle was a great solution. If you can keep your buffs from falling into enemy hands you're doing your job. You don't need to counter jungle ( try learning to counter jungle after you learn how to work your side of the jungle/how to gank ) to do your job.
The game has several types of junglers: Controllers, Farmers, Gankers and Hybrids. I'll post a quote on each type from the LoL Wikia and then post my own thoughts.
Control junglers are aimed specifically at defeating other junglers and ensuring that they have as little influence as possible through the course of a game, and assist their allies mainly though use of objective control, such as using early takedowns of the Dragon and the two major buff camps on both sides. This type of jungler can be very varied in their design but most of them will prioritize sustainability or dueling potential as their key stats, allowing them to defeat other junglers through both attrition and raw power if needs be.
Control junglers are counterpicks to ganking junglers, as their stalking nature allows them to nullify their opponent's strengths by ensuring their ganks do not succeed, as well as crippling what little farm they already get.
Examples of control junglers include Nunu and Trundle.
Nunu's Consume ( and Cho'Gath's Feast ) can pair with Smite to make an objective very hard to take. Nunu has a steroid for attack/movement speed that can help your ally run to safety when ganked and Nunu ( along with Cho'Gath ) both have CC that can stall a gank. When Nunu combines his CC with his steroid it can sometimes be enough to turn a gank into your favor, allowing you to pick off an enemy for a counter-gank. I'm also going to leave you this link so you can see an example of what a control jungler does and some ( both champion-specific and generalized ) mistakes made.
A farming jungler does what their title implies - they spend the vast majority of their time farming the camps that respawn in the jungle, and will also often supplement this income by entering an ally's lane and farming enemy minions as well. This type of jungler sacrifices aiding their laners in the early game with the intent of getting large amounts of gold and levels to become a significant threat in the late game, and often boast a very high damage output and strong scaling, allowing them to farm the jungle at maximum speed and effectively become a second carry if they are allowed to farm for long enough.
Farmer junglers tend to resist control junglers, who more often than not will simply not be able to steal camps at a rate that is able to compete with the high clearing speed of their opponent and will usually end up outscaled by their better farmed opponent.
Examples of farming junglers include Udyr and Master Yi.
A lot of farming junglers will take Devourer but if you are playing somebody more tanky it might be worth taking Cinderhulk. The smite/enchantment you take for your smite should depend on your jungler and your situation. If you are Master Yi or Shyvana you might want to take chilling smite while Master Yi will often take Devourer for his passive/q while a Shyvana might grab Cinderhulk. This changes for Master Yi or other Devourer-based if they falls behind as the further behind they are the less likely they would be to finish Devourer in an effective time-frame and thus might opt to take the Warrior enchantment instead.
A ganking jungler aims to contribute to his team by providing constant pressure to enemy laners and setting up kills for their allies, allowing them to snowball their way to victory even as the ganker jungler falls behind in gold due to generally low farm. Such junglers often tend to prioritize heavy crowd control and mobility in their kit, which allows them to reach even the most heavily-entrenched laners and lock them down long enough to ensure their demise.
Ganker junglers tend to be good picks against farming junglers, as without the need to worry about an opponent intercepting a gank attempt they can freely roam and harass opposing laners with no fear of reprisal.
Examples of ganking junglers include Jarvan IV and Nautilus.
All I have to ask is this: Please make sure the buffs on your side of the jungle do not fall into the enemies hands. If you allow this to happen you've failed because the enemy now has two more buffs they can use against you. If you see you have a buff up don't start a gank but instead get the buff or pawn it off to a team mate so you can gank without having to worry ( as much ) about the buff falling into the wrong hands.
I thought I'd give you some examples of Jungling that you could look at as needed; I hope they help.
Detailed Amumu game-play break-down: https://www.youtube.com/watch?v=QiKMuoRfLrE
Lee Sin jungle featuring live commentary from Huzzy: https://www.youtube.com/watch?v=U43_tdj_41g
Don't forget the link posted earlier! https://www.youtube.com/watch?v=b2r19BVjdkM
You can also check out this!
Detailed Vi game-play break-down; both Devourer Vi and traditional Vi: https://www.youtube.com/watch?v=rBccR_1MHLg and https://www.youtube.com/watch?v=yZ8Y4zYv-oA
I hope all of this helps!