Howdy! This is a general build and guide, so I'll try to keep it informative and brief. So, welcome to Annie, The Hottest Little Girl!

Pros
Massive Burst
Good Lane Dominance (against most champions
Great Utility (Passive Stun)
Standard AP Carry that everyone should learn to play

Cons
Focused down early (like most AP Carries)
Been nerfed recently, but still very viable
Has trouble with beefy characters (ie. tanks, off-tanks, health stackers)

Ok, the important runes here are the marks (reds) and glyphs (blues). On Annie, the seals (yellows) are pretty much whatever your personal preference is. I prefer some mana regen on my AP (ability power) Carries, as none of my later items will have any, but feel free to use whatever you want there. The marks, however, have to be magic penetration. Annie's an AP Carry. End of discussion. Third, I take the ability power per level glyphs, because the item order I use does not put a lot of emphasis on AP until later in the game, and these help keep Annie's power up. For Quintessences (purples), I take one magic pen quint to give an even 10 magic pen for runes. This, combined with magic pen boots and masteries, gives Annie a cool 30 magic pen and 10% penetration, a solid amount to melt squishies and good enough to hurt the tanks until you can buy your late game items. The other two quints are flat ability power to help Annie's early game and to support my AP light item build.

Masteries are pretty standard for AP Carries. Offensive tree is pretty self-explanatory, pick all choices that increase Annie's damage output. The other 9 points, feel free to experiment with. I prefer the little extra movement speed, the reduced flash cooldown, and most importantly, the neutral buff duration increase. Quite useful.

Ok, in general, your core build is boots, Rod of Ages (RoA), and Rabadon's Death Cap and/or Doran's ring. I like to start with Doran's ring, but you are more than welcome to do boots and 3 pots combo. Both work, and the second one saves you a wee bit of money, because you'll eventually sell the Doran's ring. Either way, build RoA as soon as you can, starting with the catalyst for lane staying power. After your core build, the next three items are situational. If you're getting focused hard, grab Rylai's to beef yourself up and slow your enemies down. Having trouble with the tanks, grab the Void Staff. Abyssal Scepter is usually the last item I build, and I prefer it to Zhonya's for two reasons. One, Abyssal Scepter + Void STaff + other magic pen bonuses pretty much give you true damage on squishies and nasty amounts of damage on tanks. Two, with Annie's "E" skill (Molten Shield), your armor and magic resist will both be around 130, combined with it's damage returning shield and approximately 3.2k health, you become much harder to kill. That being said, Zhonya's is perfectly fine, especially if the other team is AD heavy.

Ok, the skill sequence is kind of important. I know some people preach maxing the "Q" (Disintegrate) skill first, but I disagree. I start with Q, and use it for the rest of lane phase for two purposes only. One, last hitting minions so that your mana is refunded, and two, to stun the enemy champion so you can open up your combos. Annie's "W" (Incinerate) skill does much greater damage after putting 2 or 3 skill points in, and most players, especially good ones, will not allow you to hit a Q, W, Q combo unless they suck. If they suck, then it doesn't really matter what your skill sequence is. However, lets assume they don't suck, and with a maxed "W", a stun combo of Q + W let's you get in, do massive damage without over extending or leaving yourself open to a counter attack, and then start re-stacking your stun. One of these combo's, combined with a Ult + W + Q Ignite combo as soon as your stun is up again will kill most champions, and if not, it will at least force them to leave lane.

Well this is just a pretty standard AP Carry set up, and for Annie in particular, these two are pretty much the ones you have to get. Flash + Ult combo, very deadly.

This isn't supposed to be an extensive Annie play-style guide, however I will fill this space out with more playing tips if I some comments asking for such. Otherwise, just a basic overview for Annie. Your job, generally speaking, is to burn squishies (AD or AP Carries, sometimes support chars). Fortunately, this build allows you to do that spectacularly well. Also, do NOT waste your Ult + Stun combo on beefy chars. That's bad. If you feel the urge to start wacking on tanks, and you seem unable to reach the squishes, then hit him with a "W" or "Q" and just bring your stun back up. Always, always, always hit as many squishies as you can with your UltStun in a team fight. Side Note, try to stun AD Carries particularly, because lifesteal sucks. This means a victory, and victories are good. Lastly, always have your "E" on. It costs essentially zero mana, because you'll have regained the mana you spent on it by the time the cooldown resets. In a team fight, that E turns you from an AP Carry to a 'Ow, this little girl burns and won't die' AP Carry; which, obviously, is better.

Before wrapping up the guide, I'm just going to make a few points that I think most people will bring up. First, why no defensive items (ie. Banshee's Veil)? Well, the explanation is fairly simple. You are an AP Carry, and every item chosen in this build reflects the need to have more AP to do more damage. That being said, the combination of RoA, Rylai's, and an Abyssal Scepter (or Zhonya's) with Molten shield make Annie fairly beefy for an AP Carry. And by fairly beefy, I mean 'Wow, I didn't really buy any defensive items, and I'm still pretty hard to knock out'. Now, Banshee's Veil has it's place, and every game is different, so you make adjustments as needed. But, in GENERAL, you are going to wanna go with the damage output, as that will be the role you are fulfilling on your team.
Second, I'm sure some people will want to buy Mejai's, and my response is 'No'. Never. Don't waste your money. No matter how well you are doing, you will be focused down, and you will die, and the advantage gained by having Mejai's will be null. It takes up space for better, more valuable items and also begs teams to focus you even more to keep your stacks down. Don't buy it. Bad. No.

Ok, so them's the basics. Feel free to comment, as I will read them. If some parts were not explained clearly or well enough, I'll edit the guide. If there are differing opinions on items, masteries, runes, whatever, I'll do my best to explain why I choose one over the other. If a comments suggest ideas that I like, I will take those ideas and pretend they were my own. Thanks ;)