Sion was the first character I ever bought in LoL, but I didn't really realise his full potential until much later. I built him as a full tank when I first started, but there are better full tanks. I tried him AD, but there are better AD's. I then decided to try him as an AP Burst mage, and fell in love with his insane burst and tankiness. If he gets fed, he is essentially unbeatable, and he soon became my main character.

I decided to do this build because, while there are a few AP Sion builds out there, all of them focus on his Q and his W, and buffing up his AP. While they are useful, they don't get the best out of him, as they make his E and his ult almost useless. I wanted to make a better rounded build, that takes a bit of hit in AP, but lets you use your E and your ult, while still retaining Sions awesome burst power and tanky ability.

Pros

Great burst damage.
Really tanky stats to go with his insane damage.
Awesome ult.
Stun.
Insane shield.
Underplayed, so not people underrate you.

Cons

His E is only useful when ult is popped, or when farming.
If they burn through your shield before you pop it, there's a long wait.
A bit farm/feed dependent.
He's slow.
Mana-hog early.

Your items are the most important part of AP Sion. Build him wrong and you'll cripple him. Build him right, and you'll find it very hard to lose.

Always, always, always rush a Catalyst. I don't care what your feed is, don't be tempted by stack items like Mejai's. An early Catalyst is your best friend, as it refreshes health and mana, which is really useful for early game Sion, who's very mana dependent.
Then you move on to your Sorcerers shoes. These give you your mag pen, which is quite useful if you've killed a few and your enemies start stacking mag resist.

After these two, start moving into your core build. This is-
Rod of Ages; get this first, because it stacks with time. Fully stacked, it's a force to be reckoned with, and also gives you a lot of tanky ability.
Morello's Evil Tome; the CDR reduction on this is pivotal with Sion, and will let you spam your Q a lot more.
Lich Bane; gives you AP, a bit of tankiness, mana, and an awesome passive which is really useful for your stun, auto attack, shield pop combo.

After that, you can start adapting depending on situation. I like to build a Nashors Tooth and a Hextech Gunblade, because they not only give you some AP, but they also help out your other abilities with the Attack Speed bonus and the Lifesteal, Spell vamp and extra AD.

However, if your in a situation where you're either the sole AP character, up against a tanky team, or just want extra firing power, then ditch the Lich Bane, and build a Deathfire Grasp (for the CDR reduction and active), a Rabadons (AP FTW) and an Abyssal Sceptre (bit of tank, mag resist reduction). This is a full AP build however, so don't expect to get much use out of your E or ult. (Seen in second build)

A basic breakdown of your skills-
Q-
This is your spammer. Fully built, with CDR runes and items, you should be able to get it off a few times every fight. If you get fed, it'll take about a fifth of their HP and hit them with a massive stun.
W-
This is your oh-****-what-just-happened spell. Fully built, you'll be hitting *****es for about half health. Activate it before you get into a fight, and then stun, auto-attack and pop. It's extra fun if they're not familiar with Sion, as they'll just think it's a shield. It's also got a fair bit of range on it, so if someones running away, it's a good way to **** them up, and is awesome for destroying minion waves for some extra cash.
E-
Your money spell. Make sure you get it early, when you're still last hitting with auto-attacks, because the passive is really useful. Also great in tandem with your ult, or if an enemey is really low and your spells are on cooldown.
R-
Literally a game changer. Pop it in team fights and you'll make your team invincible. Has a super low cooldown for an ult as well (lasts 20s, cools down in 60s), so it can be used to push towers, or just as an extra health pack. Also great if you want to get a cheeky Baron.

Your main combo with these spells should be activating your shield, running in, stunning, auto attack once or twice, shield pop, then, depending on how fed you are, finish them off or back out. With this combination, you should be able to burst down most enemies from full to about a third health.

I normally take Sion with Exhaust and either Flash, Ghost, Ignite or Clarity. Exhaust is great because you don't really get a whole lot of movement speed out of him, and it makes 1v1s really easy because of the damage reduction. What you take with it depends on play style. Flash is useful to catch up with faster enemies if your shield is about to pop. Ghost is good to initiate battles, as you can jump in, stun, shield pop and then get back out again. Ignite is good for the heal reduction, and as a last resort killer. Clarity is great for early game Sion, as he can be a bit of a mana hog if you spam his spells, so it can keep you in the game.

One of the most important parts of playing an AP Sion is getting a good minion farm. You're pretty much totally item dependent, so in order to get that ridiculous burst damage, you're gonna need some gold. It's hard to get creeps early, so hang back and only go forward for an easy creep kill. Once you hit level 4, you'll have your E up, so be sure to toggle that on and off for last hits to get that +1 HP. Once you've started moving into team fights, you should be around level 8-10, and you should have an average farm. Now is when you really need to start getting that minion score up. Whenever there's a break in team fights, go into an unoccupied lane, walk right into a big group of minions, and pop your shield. You should be getting whole waves at a time after a while, which will bring your farm right up, and let you get those more expensive items.