I mainly use this build that makes Brand a very powerful champion.
You have to be a little careful of your mana because you won't have much, so you should get the blue golem all the time or just use spells when you have to.

Runes are not a huge deal, for AP champs you should always get AP runes and MP runes, the latter being the most important. You can even throw some CD-red runes in the mix or some mana if you will.

This masteries sequence works for all mana-based AP champions, it's one of the most affective.

With items you can try different paths, but I for one use a path with low mana and alot of damage output.
First i get a doran's ring because it's good to start with, or if you want to skip that item get the tome 1st and mana potion.
When you have about the amount of gold for the boots go get them, then build to Rylai's which is one of the most important items for Brand. All the things he needs to be a sturdy nuker (some HP and AP and that great slow that makes it easier to hit them with all your spells, ulti can reach them easily, you can better put the pillar of flame and so on...).
And after Rylai you can get basically any AP/MP items, but I strongly suggest getting Rabadon's and Will of the Ancients.

1st of all max out the Pillar of flame and Sear because these 2 do the most damage and it's great for stuns or making use of the 25%+dmg on the pillar.
If you fight vs 1 enemy use conflagration on him and then pillar, then sear, this will break him down.
If you fight multiple enemies use pillar 1st then conflagrate them to spread the fire and then use pyroclasm which will devastate them.
Rinse and repeat. :D

You can chose between: Flash,Ghost,Ignite,Teleport,Clarity.
I mainly use Ghost and Ignite because these 2 are very useful.
Ignite for tanks and for early game to have more damage.
Ghost to run out of or into fights, or chase some champion.
Flash could be useful too for escaping mostly, Teleport is great early when you have to go to base for a refill and enemies are banging the tower, Clarity is also good because of the mana problem.

Your job is to not get targeted, stay behind and unleash hell upon them.
Don't be the hero and don't be the one who gets in 1st!
Have patience and you will transform them to charred remains in seconds after the fight begins.
Having a mass stunner or disabler in your team is very useful, Amumu, Kennen, Nunu, Galio, Malphite and other champions that can make the enemies stay in one place for a bit.

So this is about it, play Brand carefully, he's somewhat squishy.
Have fun burning down your opponents like little flies!