Not Updated For Current Season
The masteries shown here are not yet updated for the current season, the guide author needs to set up the new masteries. As such, they will be different than the masteries you see in-game.
I get the Amplifying Tome first in order to get those early kills that would otherwise be missed without any AP. I follow up with the Sorcerer's Shoes to get some magic pen (Brand does fine without the CD reduc) and the Tear Of The Goddess to build up your mana for when you finally get Archangel's Staff. I like to build a Will Of The Ancients (if you insist on CD reduc, switch this out for Morello's Evil Tome. If you happen to be a good player and looking up guides for some reason, I get my Majai's Soulstealer here but I assume that player's looking for help wont utilize it fully.) before AA for the spell vamp, but it works either way. Then Rabadon's Deathcap, obviously for it's massive AP boost. Here I would get either Zonya's Hourglass or Abyssal Scepter depending on what kind of damage you're taking more of, followed by Rylai's Crystal Scepter for the slowing effect and the health/AP. Keep in mind that the calculations above do not include Archangel's and Rabadon's passives. This should get you ~750AP.
I Prioritize Pillar Of Fire because it's a great harassing ability and easy to earn CS with. It's a good idea to get one level in both of your other abilities in order for both your Champ slaying combo (Conflagration-> Sear-> Pillar Of Fire), and your AOE combo (Sear-> Conflagration-> Pillar Of Fire) to be used. After maxing out Pillar Of Fire I favor Conflagration because (to me at least) Sear is far more useful as a stun than a damage dealing technique.
Ghost/Flash is the combination I use because Brand is majorly lacking in survivability and this makes up for it. Taking ignite is ridiculous seeing as your passive is very Ignite-like in it's nature. I could see Exhaust/Flash or Teleport/Flash as possible combinations, but this is my preference.
They give you AP and Magic Pen, I never really have mana issues and CD isn't too necessary on Brand, seeing as my Ult usually ends team fights and his other abilities are really quickly recovered.