Nunu's job is to make the enemy jungler/laner feel miserable. He is an amazing counter-jungler and basically, despite not being the champion with most to offer (damage-wise and low outplay potential), he can effectively turn games into 5v4 if he makes an early game jungler fall behind before doing much. Another thing he excels at, is securing buffs, with that fatal weakness that is not having anything to go over walls. However, a flash-smite-consume steal is still quite beautiful to pull off.
Nunu's passive is what lets him start red instead of blue, which is quite powerful considering the mana problems some junglers are having after the regen nerfs. I don't really think it is powerful enough to be itemized for, however, but a Wit's End can still let you drain some MR and make your snowballs deal more dmg. So I don't think it is that bad of a choice.
Nunu's Q is amazing for a jungler, or even a laner, taking in account its reduced cooldown and the fact that it heals you even if it won't kill the minion (Cho'Gath can't brag about this). This also lets you beat minion-mancers like Heimerdinger or Elise and grants you decent buffs from eating different monsters, which can all be possessed at the same time with little to no CDR.
Nunu's W is what keeps him good late game. If a Vayne gets her hands on this buff, she will deal a ton of true damage, while being extremelly hard to catch. Or, if your adc is behind, you can always give this to a slower tank that wants to engage (Voli, Udyr, MS based engagers that aren't in the best spot right now). Not bad, as it also lets you move through the jungle and from your base faster, and lets you stack your passive more quickly as well, but it still feels a little lacking, somewhere.
Nunu's E is my favorite skill. This makes him a huge lane bully with the slightest bit of AP, and this is what lets him have an almost unparalleled burst, at the expense of tankiness and the ability to steal anything since you will blow up before getting close to the objectives. This is also quite good for peeling for your carry, and for dueling yourself, since it slows enemy's movement and attack speed. Put this on a Tryndamere while giving your W to your carry and he will have a very hard time killing them. This is also your primary ganking tool. Your ganks aren't that strong, but with a Red or a Chilling Smite, it can happen. This is your only chance to avoid getting kited, which makes a Banshee's Veil a good counter to Nunu in general.
Nunu's R is the easiest channel in the game to cancel, unfortunately. This forces you to play smart. You must wait until the enemy blows all their CC, but so that they aren't low enough to flee. Then, you must rely on your W to get to the middle of the fight to begin channeling, because you aren't Katarina, and you can't just blink into it. If you pull this off, the enemy is either running away or trying to kill you. When you die, the damage is applied, so there is always hope until the end. Since this also slows enemy MS and AS, this is also great for shutting down any AA reliant champion that isn't Yi. And the burst on this thing is huge too! For ganking, this skill is not your best shot, but if you get on their escape path, they are most likely flash-less or dead.
Cons:
No hard CC
Either tanky and damageless, or squishy with damage (Lee Sin, for example, can be a mix of both very well)
No hard mobility
Very vulnerable to being interrupted
Falls off damage-wise
Low outplay potential
Items for tankiness:
The slows make you an optimal chaser, as well as a good shutdown for AA reliant champions. The visage is for AP comps and to let you heal off more with your Q if you took some damage while walking up to the objective you were about to steal.
Items for burst:
There isn't much to explain. Pure AP and Penetration. I don't recommend building all of these in the same game, unless you are very far ahead or playing him as an APC.
Items that mix a little bit of both:
I am not a fan of Rylai's on Nunu since your offensive abilities already slow by themselves, but if you need HP and AP, I guess you can build it. Rod is usually a good choice in most games, but the other two are kind of situational as well. If any of these, I would pick Rod above the others.
Other relevant items:
- Never thought of it, but it might be actually good if you can stick to your target
- Strong ganks, or mildly strong. You will need it if you want your team to snowball.
- I know this could go to the burst category, but I don't think of it as a very reliable choice. If you have some AP however, the MS it gives can help you reach your target and proc the Spellblade. Think carefully when buying this item, as I don't think it is that great.
- I still feel like Rod of Ages outshines this most of the time, since your W is a good replacement for the active, if you want it for more chasing power, who am I to judge?
- I don't know... Warmog's feels a little outshined to me right now... But it is still a solid item if they have true damage in their team.
