Sorry if this build lacks some of the elegance that others have with pictures ect. I designed this guide to show others how I play Brand. Unlike other builds, this one is designed around building utility such as Rylai's Crystal Scepter and Abyssal Mask, as his poor Ability Power scalings make building straight ability power a waste.

For Summoner Spells I take Ignite and Flash. Ignite has great synergy with Brand's Passive, ensuring early kills in lane. Because Brand is a hard-hitting AP carry, he is often targeted in team fights, thus I take Flash for the quick escape. Although flash has recently been nerfed, it is still a viable summoner spells, and can be used for escaping dangerous encounters, as well as offensively for securing kills.

Alternative Summoner Spells:

Ghost: Great for escaping and chasing. Personally I find Flash more effective, but if Ghost works for you then by all means use it.

Teleport: Great for getting to and from lanes quickly. An overall useful spell.

Cleanse: Always a great summoner spell to use. Great if the other team is focusing you with a lot of Crowd Control.

Clarity: If spamming abilities and running out of mana is your thing, then Clarity is perfect for you.

Heal: Useful if laning extremely aggressively. Allows your champion to dive under turrets and win more 1v1 skirmishes.

For Runes I take standard AP caster runes. I take Magic Penetration Marks and flat mana regeneration Seals. I also take flat magic resist Glyphs to provide defense against other AP casters who go to mid lane (i.e. Le Blanc, Malzahar). I grab movement Quints for the maneuverability, as I find they give me more maneuverability as well as making kiting spells easy. (Alternatives include Health Quints for sustain, or Ability Power Quints for more early game damage)

Note: Recently as the meta game has shifted and Blue Buff is normally always given to the mid laner, I swap out mana regeneration seals for flat armor.

For skills, I start with (1) point in Pillar of Flame at Level 1, (1) point in Conflagration at Level 2, and (1) point in Sear at Level 3. I max out Pillar of Flame immediately, as this is his most useful spell for dealing massive Area of Effect damage to champions or minions. Remember that this has the longest cooldown out of all of your spells. I then max Conflagration as it helps both with harassing enemy champions and farming. Conflagration is a great harass tool and can also be used to secure kills. I max out Sear last, as it is only useful for it's stun and poking at enemies who play defensive. Upgrade your Ultimate Pyroclasm whenever it becomes available.

Edit: Against passive laning opponents, it may be wiser to max Conflagration first over Pillar of Flame. This ensures safe damage and poke as well as an easy way to proc damage from your passive.

Always start with Boots when laning mid. This allows you greater mobility and harass in lane. Health Potion's give you great sustain in lane, so always grab some if needed.

As soon as possible, upgrade into Sorcerer's Shoes for the enhanced movement speed and magic penetration for his passive Blaze. I normally buy tier 2 boots after I buy (2-3) Doran's Ring. My first major item is Rylai's Crystal Scepter. This gives a large ability power increase, durability increase, and allows Brand a constant slow with Blaze. Depending on how I'm doing, I may opt to buy an Abyssal Mask if laning against an aggressive champion such as Kassadin or LeBlanc.

My next major items are Void Staff, Will of the Ancients, and Rabadon's Deathcap. These three items can be built in any order depending on the situation of the game. Void Staff gives massive magic penetration ensuring all foes die to your combo. Will of the Ancients gives brand sustain in lane and dueling ability with champs like Casseopia and Ahri. The extra aura is also very helpful for any of your teammates that can utilize the spell vamp and ability power bonus. Rabadon's Deathcap is great for an offensive boost. Even though Brand has poor Ability Power scalings, this item is still a good buy and will increase your damage output by a lot. (If your opponents start building lots of magic resist, it's good to buy Void Staff first)

I wrap up my build with Zhonya's Hourglass. This item is great on Brand because of his long cooldowns. After nuking with all of your spells, immediately cast the active, which will grand you invincibility for 2 seconds; granting time for your spells to cooldown. This item also gives a great amount of Ability power and decent Armor.

If the game progresses, I finish Will of the Ancients (If not bought earlier) as it benefits all of your teams AP champions in team fights as well as it's great synergy with your ultimate. However if you do not find a need to buy this item earlier in the game, it is suggested your support champion buys Will of the Ancients, as it frees up your item slots, or having both the support and you with a Will of the Ancients as it provides you with double the spell vamp and ability power.
Alternative Items:

Deathfire Grasp: Buy this item if enemy champions build lots of health. (i.e. Volibear) Because of the close to 700 AP achieved late game, this item will deal magic damage based on +50% of target champions health. This is a great way to initiate and tear tanks apart.

Lich Bane: This is a fun item to use on brand, as his basic attacks have a sick animation. After stunning someone with the (E-Q) Combo, proceed to humiliate them with auto attacks.

Archangel's Staff: In general I don't recommend buying this item on any champion as it is highly impractical. If you are having mana problems, Catalyst the Protector will suffice. But if you really like the way it looks in your item slot, then by all means go for it.

The standard combo any Brand player should memorize is Conflagration into Sear into Pillar of Flame. This combo is great for any 1v1 situations. Lead with Conflagration to ensure a Blaze on the target. Immediately follow with a Sear to stun the enemy, then use Pillar of Flame for massive damage. If this standard combo is used with your Ignite summoner spell, it will kill most champions even without using your ultimate.

For team fights Lead with your Pillar of Flame to deal massive Area of Effect damage, as well as to inflict a Blaze to each target. Then fire Pyroclasm to bounce between enemies dealing tons of damage. Use Conflagration and Sear to clean up any remaining enemies.

Be sure to remember that these combos are not set, and can be used differently depending on the situation.

Always remember to last hit as Brand. Even though you are able to clear minion waves extremely fast using the Pillar of Flame, Conflagration combo, resist the urge to do so as you will push your lane making you easy to gank. Conserve your mana and last hit, remembering to poke your opponent with Conflagration now and then.

Always look for opportunities to take your enemy jungler's wraiths by simply walking over to their camp and using the quick combo of Pillar of Flame into Conflagration. Be careful for wards. If your lane is pushing and you feel unsafe, you can always just as easily walk over to your own wraiths and take them if your jungler is ganking or at another camp. Remember that wraiths spawn after 50 seconds, and keep a tab on this timer as it provides your champion with a large Exp. boost as well as taking away the Exp. from your opponents.

Remember the role you play in a team. You are the hard nuker who will be focused either instantly or second. Therefore you should never lead the fight with Brand. Always in team fights, hide being your tank so you can cast spells safety. Make use of his Area of Effect damage and look for opportunities to effectively use Pyroclasm. And of course those Penta-kills at Baron are always enjoyable.

Pros: Great Area of Effect Nuker, deals massive damage to groups of enemies.
Cons: No escape method besides situational stun, squishy, long cooldowns.

How my build counters these problems:
-I take Flash as a summoner spell for an effective escape method.
-I build Zhonya's Hourglass to counter his long cooldowns, using it's active to allow for some spells to cooldown.
-I build Rylai's Crystal Scepter which gives more HP and durability, as well as a very strong slow with his passive ability.
-Late game, Will of the Ancients can be a viable item for keeping you and fellow AP teammates alive.