Welcome to my guide for Brand, the latest champion to join League of Legends.
Brand is a combo-based AP caster, relying on chaining his abilities together for utility and maximum damage. I've been playing Brand since the day of his release (when I can, due to servers), and i've focused on two different ways of building him that I find work the best.
The first (team 1 in this guide) is to build him as a simple AP nuker, with AP/health items to give him some bulk. This build is better for people just starting to play Brand, as you get more AP and survivability early in the game, although you don't reach as much total AP lategame.
The second (team 2 in this guide) focuses on maxing your mana early-game while laning. This allows you to harass constantly once you get your third item, which helps in laning, hurts your opponents and builds up your mana quickly. Since Brand is so combo-based and has low CD's, you can max out Tear of the Goddess's mana effect quickly. This huge mana pool is then turned into AP by Archangel's Staff. This build gets more AP lategame (you get another ~400AP from the dual Archangel's Staff, but is harder to play as you are easier to kill. Brand has already high base damage on his abilities, so you should be able to get kills anyway, but it requires a different playstyle.

Since Brand is an AP caster, runes are pretty simple. If you don't have these particular runes, then take what you have, but these are I've found work well.

Pretty self explanatory, more MPen = more damage. Also, the damage from your passive Blaze is increased by this, so it's win-win.
Greater Seal of Replenishment
More important for the second build, but I take these for both as Brand is pretty mana-hungry. Allows you harass plenty early.

More skill spamming = more damage, longer combos, keep Blaze on your enemy 24/7 :)
Also allows faster mana charging for the second build.

MORE magic penetration, same as marks. I would normally take AP quints for casters, but MPen synergises so well with Brand's passive :)

Again, pretty typical AP caster masteries, focusing on CDR, MPEn and mana / mana-regen.
9-0-21, taking 9 in Offensve for the CDR and 21 in Utility for the CDR, mana regen, mana and the summoner spell cd reduction.
The main ones you want points in:
Offense:
Sorcery, Archaic Knowledge
Defense: N/A
Utility:
Perseverance, Expanded Mind, Meditation, Intelligence, Presence of the Master.

Fairly typical AP caster with a bit of health thrown in. You get ~500 AP and ~3.2k health, so you're a credible threat even with Brand's less-than-stellar AP ratios. However, you are primarily an early-game champion. If you play well, your opponents should be underleveled, underfarmed and should have very little gold. Your main aim should be to get your enemies to surrender, as the longer the game lasts the weaker you get compared to other characters.
Start off with Doran's Ring, as it allows you to lane longer - mana should be your main problem, not getting low health.
You should take mid lane, preferably. Solo-top isn't as viable, so if you can't get mid, go bottom with teammate. (Your team should let you go mid, if they aren't, then you shouldn't be playing with them :P )
You want to focus on last-hitting minions for gold, while preventing your opponent from last-hitting. This is the mistake that most Brand players I see make - last hitting, but failing to effectively harass their enemy. With Pillar of Flame you have an extremely effective zoning tool. Your enemy will either have to move out of the way and sacrifice some gold and exp, or get hit by it - and because we max this first, the damage is nothing to be laughed at.
Your AOE spells are also extremely potent at farming minions. Pillar of Flame -> Conflagration should leave minions low enough to kill with one auto-attack. While you should use these to farm, you need to be careful of two things - Firstly, try not to run out of mana, as the mana costs for his abilities are high, and secondly, you should avoid over-extending. If your lane is pushed beyond about ~40% you should avoid hitting minions until they return to your turret. This is especially important if your enemy team has a jungler.

If you can safely manage it, Brand is a pretty decent ganker too - A 2 second stun at Level 1 with Sear, providing the enemy has Blaze on them. You can get an easy kill with a high dps ally using Conflagration -> Sear -> Pillar of Flame.
I'll write more about combos in the "Combos" section, further in the guide.

When you can afford it or if you get really low on mana, return to base and pick up some of your items. On my first trip back, i'll typically get Catalyst the Protector and Sorcerers Boots, maybe Rod of Ages if I get a couple of kills. From then on, just buy items when you have the money from them - but make sure you're there for important stuff such as teamfights, baron, etc.

Once the game gets to the teamfights section you are still a potent threat with your AOE combos. You need to stay out of the thick of it though, abuse the range on your spells. If you get close, you will probably die, as if you're doing well the enemy team will focus you. Stay behind your tank at all times!

The playstyle for build 2 is pretty similar to build one, but with a few things to remember that I will list here. It's these extra little things that make this build harder to play.

• You need to be able to juggle farming, harassing and mana-charging effectively without running out of mana or dieing - harder than it sounds.
• You have a "weak point" in midgame - Once you've outgrown the base damage on your abilities, but before you're turned all that mana into AP. Be careful at this point. You also need to gank more to get the gold to pull your way out of this point.
• You have less survivability early-game, so you are a lot easier to kill (see point 1.)
  
  The main difference is that you should be spamming your skills as much as possible once you have Tear of the Goddess. Before that, there's no point. You WILL run out of mana a lot, and you WILL need to return to base a lot. Because of this, you NEED to make your opponent return to base more than you. To do this, you should be harassing far more aggressively - you should be able to deal plenty of damage to your enemy without taking any damage yourself. Do this by trying to keep Blaze on them constantly. With the MPen from your runes, the damage isn't to be laughed at, and if you keep it on them all the time it quickly adds up. Once they get low, go for a kill if you can get it.

Since Brand is such as combo-based champion, I've added this section to add some clarification on possible difference combos. Of course, these should all change based on situation, but these are what I find best.

For single targets, such as during laning then I usually lead with Conflagration, because it's targeted. It also gives enemies such as Fiddlesticks a chance to use his Bountiful Harvest, which is good because I then follow up with Sear. This stuns them, stopping any channels. (E.g. fiddle's Bountiful Harvest ;)), and allows you to make sure Pillar of Flame hits, aswell as dealing the 25% extra damage.

For multiple targets, I don't lead with Conflagration as I want the AOE damage from that.
Since Sear has a low CD, I sometimes lead with this into Pillar of Flames -> Conflagration.
More often, however, I start off with Pillar of Flame -> Conflagration, and save Sear for stunning someone trying to escape or trying to kill me.

Sorry for all the wall of text :)

• COMBO! :)
• Try to use the optimum combo for the situation.
• Be careful not to push with your AOE - you'll be an easy kill for their jungler.
• Make use of your passive, Blaze.
• Most importantly, have fun, and remember that practice makes perfect :)