Pros
1: Excellent champion to pick vs Double AD comps. Nautilus has natural synergy with armor items such as Frozen Heart and Iceborn Gauntlet. As you build health/armor there comes a point when the AD building champions pretty much can't kill you :)
2: ZONE CONTROL: Nautilus is like Blitzcrank: the fear of getting hit by his CC means that when you walk up to squishy targets they will run away in fear of your CC lockdown.
3: Good in siege/poke comps: This ties into reason 2 as Nautilus can create enough zone control to help his poke oriented carries walk up and do structure damage, and your CC lockdown means your team has engage.
4: Strong Utility due to your large amount of CC
5: Riptide makes Nautilus a good split pusher
6: Titan's Wrath makes Nautilus FRUSTRATING to kill in the late game (This spell puts him at almost super tank status in the late game)
7: Can be played as a top laner, support, or jungler

Cons
1: LARGE CHARACTER MODEL: Nautilus' absurdly large character model may make you feel cool, but it makes lane trading and CSing much more of a mental excersize as you NEED to plan your route. In your first couple of games, just prepare for frustrating creep block. After a few games you get more used to this, but it still sucks to have to deal with.
2: High Cooldowns: NEEDS CDR to not feel clunk
3: Can become a mana hog when split pushing/spamming riptide

As a top lane or support Nautilus I play with one goal in mind: get as tanky as possible so that I can walk into the enemy team, auto attack everyone (therefore sequentially stun everyone) and spam my Riptide so they can't leave easily. Against a standard team composition with one AD and one AP threat I build strong tank items, which means I don't like Abyssal Mask because it doesn't offer any health whereas either Banshee's Veil or Spirit Visage offer far more magic resistance and health. I also don't purchase Rod of Ages as it takes 10 minutes for this item to only give you 500 Health. I would rather build a real tank item that have to rush an item that only comes with 300 health "out of the box".
Now while you can technically jungle with Nautilus, I don't like him as a jungler because he has a low base move speed, his clear speed is slowish, and most importantly there is a wide cast of junglers that offer similar levels of CC while having superior fighting potential and superior clear times in the early game (Lee Sin, Elise, Rek Sai, Gragas are just a few that come to mind).

Doran's Ring: Has AP, HP, and mana regen: everything a level 1 Nautilus wants. don't be afraid to build 2 D. Ring's in case you don't get the snowball required to justify Rod of Ages

Rod of Ages: On paper this item has many stats that Nautilus wants: AP, HP, Mana. However I don't like building this item because this item takes 10 minutes to fully activate, and the item offers no resistances. If you get a snowball going, then go ahead pick up this item as it's good on Nautilus when you are ahead.

Turbo Chemtank: This is a decent item: it offers good HP, mana, and the unique active can help close the gaps in teamfights. For top lane, I don't think it's necessary. For support, it's good. For top lane I prefer items that make me a full tank since I can get CDR, mana, HP from other items and I can use TP so I don't necessarily need the unique active.

Iceborn Gauntlet: This is a great item as it offers armor, CDR, mana, an AOE slow field to compliment the rest of your built in CC, the AOE helps with wave manipulation, and it adds Sheen Procs which Nautilus can make great use of through his Riptide.

Sunfire Aegis: this is a great item as it offers good HP, Armor, and AOE damage great for wave manipulation or teamfight damage.

Banshee's Veil: This is my favorite magic resist item on Nautilus as this item is cheap, offers the great spell shield, and offers high magic resist.

Spirit Visage: I'm not a fan of this item on Nautilus since he has alternative options for CDR, Nautilus can find other magic resist options that offer more utility, and most importantly Nautilus doesn't have any champion specific health regen effects so the unique passive isn't utilized well. The only time I buy this item is if I am facing a team with more than 1 magic damage threat.

Warmog's Armor: While this item has many great qualities, high HP + CDR, I usually don't buy this item due to buying S. Cape, I. Gauntlet, and B. Veil. If I am facing a team with multiple magic damage threats I consider building this item.

Eye of Equinox: If am playing support I buy this item so I can get a quick Locket of Iron solari

Runic Echoes: when in the jungle I prefer this item as it streamlines many aspects from other items I don't have time to build: It gives me AP damage, movespeed. I usually go Runic Echoes -> tank boots -> HP item. Cinderhulk into tank items is certainly an option, but I just prefer Runic Echoes, try both out for yourself.

Dead Man's Plate: It offers high HP and a move speed buff. After a S. Cape and B. Veil this item compliments Nautilus' need for a move speed buff, I usually get this item.

Frozen Heart: Offers Armor, CDR, mana. I usually prefer I. Gauntlet but this item is a good edition to Nautilus' kit as well.

Guardian Angel: Makes Nautilus more of a pain in the *** to deal with. A decent option later on after he gets some HP in his build.

Ninja Tabi: While I usually buy Boots of Swiftness If I find myself against a heavy AD threat team I build these early.

Boots of Swiftness vs Mercury Treads: B.O.Swiftness are better against teams with high amount of SLOWS while M. Treads are better against high AP damage teams/ teams with high CC

Randuin's Omen: A great response to crit based ADC's if their damage starts to be come too much to handle.

Knight's Bond: This item sucks as an individual item and isn't great to buy early. If you want to buy this item, buy it later on in your build. You need selfish tankiness first otherwise you become too squishy. However when you pair this item with a partner you two can survive extended teamfights for much longer than normal circumstances.

Thornmail: I usually don't buy this item, I usually buy Randuin's Omen as a final armor item, but if there are no AP threats then I consider getting this item.

ZZ'Portal + Banner of Command: These items both fulfill the same goal: giving armor + magic resist and help Nautilus create sidewave pressure. Buying these items back to back will give Nautilus a respectable amount of resistances, even if they lack HP. Try to buy these items back to back and communicate their useage with your team so you can set up side wave pressure, then walk across the map and you can save your Teleport.

Locket of the Iron Solari: Armor + magic resist + a great shield to use in teamfights. Use this if the enemy team has a dive heavy comp or your teamfight is struggling to survive theirs.

Talisman of Ascension: Not an item I normally recommend, even in the support role. However if you are not against AP threats and looking for another armor option this can do the trick. Also it gives a teamwide move speed buff, so it is a POTENTIAL 6th item, although not the best if you want to be a FULL TANK.

Zhonya's Hourglass: This item has AP, CDR, and armor. 9 times out of 10 this item is too low on tank stats and to high a price to be considered. This is an item worth considering if you face no AP threats or want to absolutely dork around.

Abyssal Mask: Not an item I usually buy since it has no HP but if you are against a primarily AP team threats then this item can work as it gives magic resist and AP.

Trinity Force: 99% of the time this item is not good, but if you are against an all AP team and need a sheen proc and an extra source of move speed (you won't be buying D.M.Plate in these circumstances) then this item can be considered.

the most important thing about Nautilus is maximizing his CC duration. You want to stagger your CC chain: some combo of auto attack, Depth Charge, Dredge Line, and Riptide.

Initiator Combo: Depth Charge -> Auto Attack -> Spam Riptide -> save Dredge Line incase they Flash away so you can stick to them.

Instant CC: Flash -> Auto Attack -> Depth Charge -> spam Riptide -> Dredge Line