I love playing Nunu. This guide is designed for the player who already has a general idea of how to play the game. It should be noted ahead of time that playing Nunu is not for the faint of heart. He is not designed to be a player that can demolish an enemy with a 1-2-3 combo, but instead he takes some skill to play well and excels when working in combination with teammates.

The runes, mastery points, and items have been selected with a specific purpose in mind. Here is what this build will do for you:

40% CDR with runes and masteries
506 AP
Deathfire Grasp 30% Nuke
+48.55 Magic Penetration
Excellent mana regen
Very fast movement speed

With this build should be able to excel at rapidly clearing the jungle and then tearing into lanes to get ganks. By the time that the end game comes and every fight is a team fight, you should have more than enough AP and CDR to be a nuisance (positioning is important here - don't get overextended!). It is important that your teammates take responsibility to intiate and tank damage during fights, or you can be CC'd and focused down very quickly.

Pros

Very strong jungler and counter jungler with excellent sustain
Excellent ganker
Ultimate is very strong and if used properly can turn the tide of a fight
Scales well throughout the game
Strong chaser with Ice Blast and very fast movement speed

Cons

Susceptible to CC
Very difficult to use ultimate effectively
Difficult to master correct positioning to avoid being focused down

The rune setup is pretty simple.

The focus runes will help you hit the CDR cap
Replenishment runes for strong mana regen (will be very handy in jungle)
The marks will give you a boost to your magic penetration.
Finally, the quints add another boost to movement speed which will really help with ganking and fights throughout the game. Combined with masteries, they add 6.5% movement speed which is a very nice boost.

9 in Defense in order to help your early game jungle out.
21 in Utility to help you with roaming and ganks.

Summoner Spells:
Smite - For Jungle use.
Flash - For ult placement, chasing, and escape.

These are almost must-haves for this build. If you feel strongly about taking different summoner spells (or if you are not level 12 yet), here are some others that can be viable:

Ignite - An good choice that gives you an extra nuke for aggressive ganking
Cleanse - May be a good pick if you are worried about getting CC'd and focused down.
Teleport - Helps get around the map faster. Also can be used well to teleport to a ward placed in bushes behind the enemy, allowing you to either drop a surprise bush ultimate, or to jump out and gank them from an unexpecting angle.
Heal - A defensive spell which has saved my life in the past. Some like it, others don't.
Ghost - If Blood Boil just isn't quite enough to chase or run away.

The build order is designed specifically to provide you with the stats you'll need to thrive in the jungle, do more than enough damage to gank effectively, and keep up your health and mana so that you do not need to go back all the time.

Personal Notes:
1. I love the Deathfire Grasp. The active damage ability that it has will give the ability to do the extra amount of damage to be dangerous. Seriously - being able to drop an enemy's health by 30% right at the start of the fight is amazing. Also, the other attributes complement him very well and provide what he needs in a first item (CDR, AP, Mana regen, etc). I've heard people say that it's range is too short to be viable, but at 650 and with Nunu's run speed, it is nearly the same as ice blast. If you complete the build, it will do %47.5 damage to a target with full health!

2. Sorc Shoes are my personal preference. Mercury's Treads are also a VERY viable alternative, especially if you're concerned about enemy CC. If you do not go with the Sorc shoes, it will be a good idea to work out what items or runes you will switch out to get the extra Magic Penetration.

Other items that are very good on Nunu:

Rod of Ages - Very nice stats overall. Provides a strong AP boost with some health and mana to make Nunu more durable. If you choose this, take it as a first item instead of the Deathfire Grasp, as it needs about 10 minutes before you will receive the full benefit from this item.
Void Staff - For extra magic pen and AP at a reasonable price. Note - This is overkill unless the enemy team is stacking MASSIVE magic resist.
Hextech SweeperHextech Sweeper - Excellent item if you're worried about enemy stealth players - NOTE: Dominion Game Mode Only
Abyssal Scepter - AP, MR, reduces enemy MR
Banshee's Veil - May be necessary if you have a difficult time landing your ultimate.

A side note regarding the BV
Many view the Banshee's Veil as a must have in order to allow Nunu to land his full duration ultimate. While this item does help significantly, learning to perfect timing and positioning (ie: Don't be the first one in or charge in right off the bat - THIS IS NOT TANK NUNU!) will likely help much more.
For example: Flash in behind the enemy at mid-fight, or pop out from the jungle behind them and ultimate. It will cause the enemies to either all focus you, or to scatter away from you. Either way, it takes their focus off of your teammates and gives them time to focus a player down.


Finally, there are the typical support items, which I won't go too far into depth on, but will list a few:
Shurelyas
Aegis of the Legion
Frozen Heart
Force of Nature

For the AP Jungle role, I wouldn't recommend these. There are better guides out there for the Support style.

Your role is to be a roaming Jungle ganker in the early and mid game, with the ultimate goal to be a contributing team player in the end game by landing good ultimate's, buffing your AD carry, and contributing damage wise.

Early game, you will be very reliant on the blue buff. The 20% CDR and mana regen are KEY, because they will allow you to spam your Ice Blast and Consume, increasing your jungle clear speed significantly.

note: it can be smart to place a ward at the enemy blue or red buff during the early game. Nunu is an excellent counter jungler, and if you catch the enemy at their buffs, you can usually steal the buffs and sometimes get a quick kill to boot. Be careful if the enemy jungler is a strong duelist, however. Better to grab the buff, ice blast and get out as fast as possible than to die.

Jungle Route
My route is fairly standard. If you feel confident in trying something else, feel free:

Start at wolves. At 1:40~ Hit the large wolf and then consume him, and run up to the blue buff. At 1:55~ Have a teammate leash the blue buff for you. Hit him down, consume and smite. This should allow you to hit level 2. Use Ice Blast as your mana allows. Next go finish the wolf camp, then wraiths, and then go to red buff. As you are fighting the red buff, Smite should just come off of cooldown. Smite him and then finish off the camp. Go kill the golems and then the wraiths. At this point you should be level 4.

The skill order can vary depending on how you are doing and how confident you are.

If you are worried about keeping your health up (Only bought 1 potion at the start) go:
QEQW. This will decrease the cooldown on consume and make it heal you more.

If you want level 2 ice blast for the extra edge while ganking and your health is sufficient go:
QEWE

After level 4, I always gank or apply pressure as I am able. It is important not to try to force a gank. If your teammate(s) in lane are very low health and the enemies are full, or if the lane is pushed up to their tower, a forced gank can easily give the enemy team one or more kills. Sometimes it can be enough to jump out of cover, hit the enemy with an ice blast and run off. It will force them to be more cautious and let up on your teammates.

After the gank, I look for more opportunities. If there are none, I go back and buy.

Note - The blue buff should respawn at around 7:20. I try to time my jungling so that I can be there as it spawns, which will allow me to continue clearing quickly without returning to regen mana.


For mid to end game - Team fights!
Many feel that Nunu tapers off towards the end of the game. If you are falling behind in levels or gold, this can happen, but if your team is not falling far behind, you will be able to scale quite well with this build as the game progresses. In team fights, don't feel like your only job is to land the perfect ultimate. Be sure to use your Deathfire Grasp right off on whatever target your team is focusing, Blood Boil your AD carry, and spam Ice Blasts (every 3.6 seconds!) from the perimeter. If you see the perfect opportunity, take it. Otherwise, I generally save Nunu's ult for better times.

Perfect Ult moments:
When your team is fleeing a fight, it is great to all stop in a bush and set up an ultimate ambush as the other team follows you in. The full duration ultimate can (and has many times) completely turn the tide!

Don't be afraid to use your ult to save your teammates lives. Many times your ult can discourage the enemy from chasing down your teammates. Even if you don't get any kills, it can be worth it to stop the enemy from finishing off your teammates.

In a bush! Any time that you know there will be enemies running by, jump in a nearby bush. I catch unaware enemy's all the time as the just happen to run by the bush that I am in. Obviously, don't sit in the bush waiting all day, though.

Well, that's it! Please feel free to provide me with feedback! This is my first guide, so please post a comment with constructive criticism about how to improve the guide. Unfortunately, I did not have a ton of time to make it super pretty, but it should be more than enough to get the prospective AP Jungle Nunu well on his way.

Have fun with the yeti, and don't forget to yell, "EMPIRE!" after getting kills with your ultimate. :)