Ok, so seeing as how this is my first guide, I hope to help out the people who take the time to read this. Brand: The Burning Vengeance... This champion is great for aoe, and single target burst damage (both covered later.) The way I play Brand is to grab enough cd reduction, so that I can proc Sear's stun with Sear. First things first, we talk about the build.

Cool Down Reduction
AP
Magic Pen

These are the essential AP carry stats.

10/0/20

Staying long enough in Offense to get Ignite's buff, swooping through Utility to get Clarity's buff. We also picked up some reduced time while dead, and some more CD reduction

Starting off with Doran's Ring gives the player a nice balance of stats to start off with, of course, later in the build, it is sold.
Fiendish Codex: Yep you guessed it MOAR AP/CD REDUCTION
Sorc. Shoes: Magic Pen, and some nice MS
Build straight into Morello's Evil Tome, don't pass go and collect $200, because the sooner this is complete the sooner you can burn through the enemy team. Ap/CD Reduction
Void Staff: Magic Pen is a nice stat!*Plus the AP
Rabadon's: AP dripping out of ever orifice!
Rylai's : Nice slow to go with all of the possible aoe/single target combo's you have.

Pillar of flame first so you can face check bushes and harrass from a extremly long range.
Conflagarate next because you want more gold at the beginning.
Sear for the stun.
I personally max out conflagarate first, because it is my opener.
I wait a while on maxing the ult because the combo is more important untill rabadon's.

Yeah, yeah... No escape mechanism... Ok seriously though, when you can stun in two moves and you have boots. What more do you need? Flash for those really fast coordinated ganks.
Point acknowledged. Ignite: Get the retreating player with 40 health left.
Clarity: More prescence in lane, especially with a support, you don't have to go back to base untill you have enough gold for Fiendish Codex, and even then you should still have alot of mana to harrass with. Ignite, with our masteries, will also give some bonus ap early on.

Aoe
Start off with a sear into the nearest target with the most health.
Throw conflagarate on him/her, then pillar of flame for max damage. ULTIMATE if there is no more than 2 minions around the champions!
Single Target
Conflagarate your main target, follow up with a sear to stun, then use Pillar of flame for max damage

One of the most versatile champions there is, as far as single target/aoe combos. One of my favorite ap carries, sorry for any mistakes I made in this build, as far as spelling and such.
Ok, Well thanks for taking time to read my first build!
Go out there and have some fun with Brand: THE BURNING VENGEANCE