Hello, my name is Stealthben, this is the first guide I have ever made so *hopefully* I won't have too many mistakes.
Brand does amazing burst damage, has a good cc, and is also a really good harasser. If built correctly Brand is a durable nuker that can run into the middle of the team, blow his load, and live to run away while his c/d's recharge.

Brand is good on any team that does not have a nuke AP on the team. Nuke champs include Annie, Leblanc(who is also an assassin, but still does nuke damage), Veigar. Some people would also classify Anivia as a nuke champion. This isn't always the case though, I have found that as long as you have at *least* one AD carry, a tank, and a support, you can have two AP nukes on a team and do just fine, at least in normal games.
My favorite team comps to play Brand with are ones where somebody else has some type of CC or slow. If I am laning with a Taric or a Sion, and they can get their stun off, my opponent is dead. As soon as you see that stun start to fly you should use your Conflag on the opponent and get into a position where you can cleanly hit with Sear for your stun, and the Pillar of Fire and watch them melt. Lanes or teams with a CC are by far my favorite to play with, landing skill shots takes no skill at all when the opponent isn't moving!

When you already have two AP champs on your team, you should probably play a tank or an AD toon. Three AP champs on the same team is way too much, tends to be squishy, and is fairly easy to counter with a FoN or BV.

Brand is a nuke, so sometimes you may not be able to get that last spell off as they are running away almost dead because of your c/d's. This is where Ignite comes into play. Being able to count on that extra damage to finish someone off as your passive drops off is not only comforting, sometimes it is imperative to landing that kill. Also the healing debuff from Ignite can save your life against those pesky Warwicks/Olafs, and Ignite completely shuts down Swain.

Flash is good for initiating/running. If you go mid you can use Flash to jump to the other side of the minions so you can combo the enemy champion and either send them back or grab your kill.

Ghost is acceptable instead of Flash. Although I don't like using it on nukers because I view Ghost as more of chasing/running spell. As a nuker chasing isn't going to do you any good because of your c/d's, and if you are in a position to where you can't get away with Flash but you can with Ghost then you are doing it wrong.

Clarity I have seen some people run with this, although with mana regen yellows you don't really have a problem with mana.

I put the 9 in offense for the magic pen, obviously good for nukers.
You may be wondering why the 7 points in defense. Honestly it is because Strength of Spirit is amazing. If you buy the mana crystal you have 495 mana at level 1. This gives you an extra 4.95 hp5 at level 1. This, along with two health pots, gives you enough regen to make it through the laning phase fairly easily, assuming you aren't being stupid and taking hits that you shouldn't be.(Credit for this idea goes to GreenTemplar, I don't know where he got the idea from, but he gave it to me.)
The points in Utility are fairly straight forward. The mana regen is always good on casters, the point in gold is placed because I couldn't think of anywhere else to throw it that would be more beneficial. Move speed and flash are both good masteries to have also.

Greater Mark of Magic Penetration Magic pen is always good! Increases the damage that you do with all of your spells.

Greater Seal of Scaling Mana Regeneration I like these because they make it so you don't have to worry about mana during the laning phase. With these runes you can harass or push to your hearts desire in the lane phase. Although if you decide to do both you can run into some issues.

Greater Glyph of Scaling Ability Power I have been playing around with these and flat cdr runes. I like ability power/lv because they allow you to hit a little harder as the game progresses while you are picking up that Catalyst of the Protector Sorcerers Boots and Rod of Ages. I generally don't get a single AP item until around 16 minutes in, so the ability power/lv runes help. Another reason not to run cdr runes is because once you hit 6-8 you start harassing a lot more, along with your farming. As I said, you can't really harass and farm at the same time. At least until you get that Rod of Ages. So having the ability to cast spells faster isn't the best, because all you are doing is making yourself have a need to go back to base more often.

I put my first point in Conflagration so I can have a guaranteed harass at level 1. Being able to deal 8% with your passive along with the damage from Conflag is amazing. After that you put a point in Sear so you can stun the enemy at level 2.

From there your priority should be Pyroclasm -> Pillar Of Fire -> Sear -> Conflagration.

Something to note is there are a few different combos that you can use as Brand. You can use Conflag to put on your passive, Blaze, so you can stun them with Sear and then destroy their health with Pillar of Fire. Or you can use sear to trigger your passive on a person, then use Conflag to spread it and Pillar of Fire with increased damage on multiple targets. These are only two of multiple different combos. If someone wants me to list them all just post(and hopefully upvote :p) and I will take the time to do that.

**NOTE**
Your passive WILL give you turret aggro. If you hit an enemy as they are running to a turret, and your passive ticks while you are in turret range, the turret WILL switch to you. I can't tell you how many times I have forgotten about this as I was harassing someone at their turret, only to walk back in range as the last tick of Blaze was falling off and have the turret hit me once or twice.

I start off with a Mana Crystal and two Health Potion. Then, depending on how the lane goes, I'll either buy Boots or Catalyst of the Protector. If you get boots first then finish Catalyst, if you get Catalyst then get Sorcerer's Shoes. After you finish your boots then you will want to finish Rod of Ages.
This is the point in the game where you can start differentiating your build. If you are having some trouble with the enemy harassing you then you can pick up the Hextech Revolver. If don't need health regen then go ahead and pick up your giants belt and build a Rylai's Crystal Scepter, the slow of course is amazing for hitting with Sear and getting a kill. Also the ability power of course makes you hit harder, while the health from Rylai's makes you pretty durable along with RoA. Sometime after you finish Rylai's you can finish your WoTA for some more spell vamp and ability power. Plus a nice buff for your allies.
After you get Rylai's you will want to look at the game. If it is going well pick up a deathcap. If the game isn't going well then you can pick up a Force of Nature or Zhonya's Hourglass depending on who is killing you.
Also you have to think about whether or not the opposing team is stacking MR. If everyone on the other team has a BV then you will want to have something to counter that, namely a Void Staff or Abyssal Mask
At the end of the game you should have something along the lines of Rod of Ages, Sorcerer's Shoes, Rylai's Crystal Scepter, Will of the Ancients, Rabadon's Deathcap, Force of Nature or Zhonya's Hourglass or Void Staff or Abyssal Mask

Pros:
Deals amazing damage, and is a really good harasser.
Is good in the laning phase.
Deals good damage mid-late game.
Has a stun, if you can land it.
Has an ult that is a lot of fun to watch follow people.
Has a nice range on most of his abilities.

Cons:
As with most AP nukes, if you get focused you are screwed.
If you can't hit skill shots, then you will have problems.

Brand is an amazingly fun champion to play if you can play him correctly. He can and will carry games, especially if you can destroy in the lane phase. Hopefully this guide will help give you an idea of what to build and a skill build to follow. Brand is fairly straightforward, but I thought I would throw my opinion out there! Thanks for reading this :).