Let's cut to the chase. Nautilus is the most useful champion on the game. You heard me. Every team will want you on it, and even if you feed or fall way behind or go afk for 20 minutes, you can still maintain an incredibly helpful presence for your team. This is because Nautilus is the master of crowd control. Every single one of his abilities has some sort of CC. Every single one. And the best thing about CC is that you can never have enough! You can have too much ad or ap, but you can never have enough CC. With that, let's start the guide.

PROS
+Absolutely insane crowd control. Most CC in the game
+Super useful to any team
+Stops burst champions
+Great shield can allow him to turret dive and escape
+Gets very very tanky late game, next to unkillable
+Good in nearly all phases of game
+Good engage with Q
+Blocks lots of damage if in front of his team mates
+His ult auto-chases it's target, so they literally cannot escape it
+Best dance in the game

CONS
-kind of supporty. He benefits the team rather than himself, so often you will have more deaths than kills.
-Doesn't become tanky until a little later
-Expensive build
-Can't really do much without team mates to support him
-Really really slow

So for jungling you're gonna wanna start Krugs first. Smite them. A lot of people say to start Gromp if it is near bot lane. I don't recommend this. As Nautilus, you want to capitalize on your disablers and CC. The Krugs give you a stun, which is just more CC. After krugs, run over to wolves and take those. Don't smite the wolves, and don't use a health pot. After wolves, you'll be pretty low, so pop a health pot and go over to red buff. Your smite should be up, so you should use that to get some HP back when fighting red. After that, run back over to blue. You might have to wait a couple seconds before your smite is up, but that's ok. After you have both buffs and you think you have enough health, it's time to gank.

Ok, this is what Nautilus does best. He is built for the team. However, he needs the team to help him. It's a nice cycle.

GANKING
Nautilus ganks are fairly simple. Come in behind the lane you want to gank. Run up to the victims. Pop your E (shockwave) on them. This will apply a nice slow. After that, just hit them with an auto attack and this will stop them for about 0.5 seconds. The laner you're ganking for should be here by now and able to lay down some damage. If the victim flashes or somehow gets away, try to Q them before they get to a turret. If they get under turret and you need to dive, pop your W (your shield) and either Q into them or pop you ult. This is all very complicated. Often ganking isn't as black and white as this. Basically, just gank normally, except save your Q and R for the last moments.

TEAMFIGHTS
In team fights you are a tank and the engager. Engage with your Q, and try to hit the adc/mid laner/whoever needs to die. You need to pop your W (shield) before this, so you don't take too much damage. This will help you get to the center of their team, which is where Nautilus excells. Once close to everyone, E them with your shockwave and if your shield isn't dead yet just autoattack them. Your shield provides a nice passive which makes your autoattacks do area of effect damage. Your ult should be saved for if anyone is trying to get away, or if you need to disable the adc for a few seconds. Your ult is best used on whoever is in the back, usually a ranged champ like the adc or apc. This is because your ult will knock up the whoever is between you and them, so you could potentially disable their entire team.
Nautilus coming out of the jungle to urf up the enemy adc