Annie is one of the most flexible mages in League of Legends, often played as a support, burst carry, or magic DPS champion. On occasion, she can even make a decent off-tank. She's most commonly played in the middle lane, and as a bottom lane support, but can fare decently in the top lane as well, in a number of match-ups (particularly with a DPS build). For this guide, I will be covering solo lane builds, and not support Annie.

In mid lane, usually take Flash + Ignite. Can opt for Flash + Ghost, or in rare cases, Exhaust.
For top lane, Flash + Teleport is often the way to go, although the other options are still sometimes viable, particularly Flash + Ghost.

Items:
Starting:
Doran's Ring gives you mana sustain, health, and threat for pressuring your lane early.
Corrupting Potion is better with Teleport in top lane, but gives you mana and health sustain, and some added damage. Care low health pool.
Dark Seal is your greed option. Don't build it unless you can make it pay off. Can also pick one up if you get first blood and feel you can snowball.
Burst Core:
Morellonomicon while not core to dealing burst with Annie, it scales well, and allows incredible lane pressure for farming and snowballing. Can sometimes opt to go 2x or 3x Doran's Rings straight into Sorcerer's Shoes and Luden's Tempest for a faster burst spike, but less pressure, and a bit of a mid-game dip.
Sorcerer's Shoes are your burst shoes of choice. Can build Ionian Boots of Lucidity instead, for pressure, or against someone building magic resist early. Will hurt burst, but help laning pressure.
Luden's Tempest is your 1-shot button. Basically enables you to kill people with your ult combo in one hit, most of the time.
Void Staff is necessary for helping you deal AoE in team fights, once magic resist items start being completed.
Other Items:
Zhonya's Hourglass is pretty much mandatory on burst mages like Annie. It allows her to initiate (with Flash sometimes), get big AoE stuns with Summon: Tibbers + Incinerate, and then stay alive for the next two seconds, while her team catches up to her in the fight. It can also be used to dodge damage that would otherwise take her out of a fight.
Rabadon's Deathcap is Annie's big damage multiplier late game, once the rest of her core is done. If all you need is more raw damage, it's usually your best option.
Ionian Boots of Lucidity: are great for lane pressure, and for dishing out DPS, rather than burst.
Abyssal Mask is a good option if you're trying to burst another mage before they can burst or DPS you.
Liandry's Torment is a great tank buster item on Annie, later in the game. Also provides a lot of AoE DPS, as Summon: Tibbers will constantly proc it on nearby targets.
Rod of Ages is a good first item on Annie when going for more of a pressure laning style, rather than burst. It allows her to trade health for damage more safely, and be more of a consistent bully, rather than burst threat.
Rylai's Crystal Scepter is particularly good on a DPS build, mid game, in conjunction with Liandry's Torment.
Guardian Angel is a moderate late game option for a DPS build, if you don't need more damage.