Brand is an excellent character to buy and I would recommend him to anyone who is capable of playing both defensively and offensively, as well as having the amount of knowledge and skill to know when and how to poke. This guide focuses AP. This is how I nearly always play him, and I win roughly 8/10 games with him. Note that you should almost always play mid as Brand.
I choose scaling runes because Brand is already OP enough early game that you don't need the edge of flat AP runes. It all adds up in the end, especially with the extra 35% AP increase from the Deathcap and your Mastery Page.
You always want to get your Pillar of Flame first, no matter what. It is more mana hungry than your other skills, but it is great for poking and farming. After your Pillar, you will want Sear for its stun. You are already deadly by level 2 with this combo. Get some auto attacks on the champion and, when they are at about half health, flash past the minions, use your Pillar, use your Sear, ignite, and auto attack. They may try to run away after that. Depending on your health, either retreat or follow them and auto attack to be sure you get the kill.
You want Conflagration third, just because it gives you a chance to stun them for less mana than Pillar and Sear. It is also great for farming when combined with Pillar. The next few levels of your skills should be played situationally. If you are having a hard time hitting your enemy (i.e. they keep dodging your Pillar of Flame), build up your Conflagration so that you can use your pillar on minions then, while the minion is still ablaze from your passive, use Conflagration on the minion while the enemy is close enough to the minion to also be hit by it.
I recommend steadily building up your Pillar as much as is possible because it hits multiple people for potentially massive amounts. Get your ult at level 6 because it also does massive damage. With all of your skills not on cooldown, you can most likely 1v1 anyone in the game (assuming they aren't fed and you aren't under leveled) when you are level 6.
From there, just upgrade skills as you use them the most.
Some people criticize me for not getting a Doran's Ring first, but that is just my personal preference. I see it as a waste of money as it doesn't build into anything. Brand isn't exactly slow, but there are much faster mids than him, so I get boots and two mana potions first. You should be able to then stay in your lane until you have enough gold for a Blasting Wand. With the blasting wand, you should have 60-70 AP, assuming you're using my masteries and runes. You then get Sorcerer's Shoes for the increased movement speed and magic penetration. At this time, I generally don't have to go back to spawn until I have around 2k gold. If you wish, stay until you have 2740 and get a Deathcap, or go back to get the Needlessly Large Rod and then again for the Deathcap. Here is where you should start building situationally.
If the other team is building magic resist, you should obviously get a Void Staff next, instead of a Rod of Ages. You would then most likely want to get the Rod of Ages next, then the Archangel's Staff, then the Crystal Scepter.
However, if they don't build any magic resist (and I've seen it often), go ahead and build your Rod of Ages. This, followed by an Archangel's Staff, will give you enough mana and health to hold your lane during enemy push attempts while your team recalls and comes back. Follow this with a Crystal Scepter and a Void Staff, and you should be unstoppable.
Brand is excellent in team fights. If both teams aren't quite going after each other and just waiting for the other to make the first engage, you can easily slip in a few uses of Pillar of Flame and back out before they can do anything. If one enemy is chasing a low health teammate, Brand can easily use either Pillar of Flame or Conflagration, quickly followed by Sear, to stun and possibly kill the enemy champion. At the end of a team fight, Brand's Pyroclasm can be used to do massive damage and possibly kill multiple enemies.
Brand is also great at ganking, provided the enemy is overextended and doesn't have wards. From mid lane, you can go up or down the river and hide in the bush. When the enemy champion you wish to gank is close enough, you can use your Pillar of Flame to get them. If you happen to miss, run in and use Conflagration (but only if you have enough health). After either of the aforementioned circumstances, use Sear to stun the enemy. You should ping the enemy when you are about to start your combo so your team knows to go in.
Brand's passive, along with ignite, can get easy kills on low health champions running behind their towers. If you see one (and both flash and ignite are not on cooldown), flash in, use your Pillar of Flame right in front of where they are going, and ignite them.
This is also good against Tryndamere when he is in his Infinite Rage state. You can use your abilities on him towards the end of this state and ignite him so that he will die when he tries to run away.
If you happen to be playing Brand on top or bottom lane, you can zone very easily, assuming you have a good team mate. Auto attack the minions and use your Pillar of Flame/Conflagration combo if the enemy gets too close.
It takes 210 mana to use Brand's Conflagration/Sear/Pillar of Flame combo. Keep that in mind.
Brand is almost always under estimated by the enemy early on. Use that to your advantage.
If you are being focused as Brand, you may want to build a Crystal Scepter earlier.