Brand has made more people scream for a nerf than I can count. Why? Because he can harass like no other, thanks to his ridiculous skill range. His ultimate, Pyroclasm, will remind people who play DoTA of Lich's ultimate in that it bounces between targets. If you are aching for massive AoEs, ridiculous Damage Over Time (DOT) passives, 2s stuns with 3.9s cooldown, and a bouncing ultimate that lands easy assists or even pentakills, then Brand might be the champion for you!

Without further ado, I present to you, Brand, the Burning Vengeance!

WARNING: This build will make you the center of every enemy carry's attention, meaning you WILL be focused down. Make sure you save your stun combo should that happens.

NOTE: This guide is meant for Summoner's Rift (5v5) maps only. If there are requests for Twisted Treeline (3v3) maps, leave a comment below.

First of all, the skill rotation:

In non-team battles,
Conflagration -> Sear -> Pillar of Flame ( Pyroclasm when there are stuff nearby to bounce off)

In team battles,
Pyroclasm -> Pillar of Flame -> Conflagration -> Sear
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OP ALERT
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At close to max CDR, Sear becomes extremely OP. Start off with Conflagration to apply your passive, then stun with Sear. With a CD of just about 3.9s, you can almost throw another Sear to stun your target after that initial 2s stun. Blaze lasts for 4s, so stunlock that nub and make him be like WTF OP!?

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Explanation:
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For non-team battles, Pyroclasm has limited effect so starting off a fight with it is not the best idea, unless your target is near a creep or another enemy hero for Pyroclasm to bounce to and fro for maximum effect.
Conflagration is your sure-hit skill that ensures you apply your passive Blaze on your target.
Since this is a non-team battle, your target would be relatively alone and unblocked. Hence, Sear right after Conflagration will stun the target for 2 seconds.
While stunned and still affected by Blaze, send your target to hell with Pillar of Flame, which deals +25% more damage when your target is affected by Blaze.

For team battles, the skill order changes. Your objective is to deal as much damage possible to as many enemy heroes as possible. And what better way to start the deadly combo, than your ultimate Pyroclasm?
Pyroclasm right into a team battle will damage most heroes relatively severely, depending on your AP, as well as applying your ridiculous Blaze passive.
With your passive activated on most enemy heroes, make them feel the heat with Pillar of Flame, causing 25% more damage to those dumb enough to clump together.
Then, use Conflagration on an enemy hero still affected by Blaze, causing it to spread to nearby enemies.
People trying to run away from you? Sear for that OP 2s stun, and attack at will.

Most, if not all will disappear into ashes before you.

"They say the world will end in fire"

Laning:

I usually go to the side lanes so I can exploit the tendency for people to clump together during ganks and use Pyroclasm to bounce off them and possibly kill them. However, the middle lane is also viable.

Harassing:

Harassing is easy, given that your Pillar of Flame has ridiculous range, as well as ridiculous AoE. Also, it may say Lv 1 Pillar of Flame does 80 magic damage, but in reality it does more than that given that the Blaze passive applies a 2% of target max HP as DOT over 4s.

Once you get Conflagration and Sear, you have a couple of choices on how to harass.
1. Pillar of Flame/ Conflagration -> Sear (for that 2s stun + a couple of auto attacks)
2. Conflagration -> Pillar of Flame (For that extra 25% magic damage from Pillar of Flame)
*3. Pillar of Flame -> Conflagration

Number 1 and 2 is pretty straightforward. 3, however, needs a little explanation.
How i do this is i would target the ranged creeps with Pillar of Flame at the back of the creep wave and hopefully catch an enemy champion in the blast too. Otherwise, it's fine. Then as they think your Pillar of Flame is on cooldown and goes forward a little in an attempt to last hit your creeps, use Conflagration on the Blaze'd creeps to spread it to him, dealing magic damage and your passive. After a few rotations, he will stop coming near his creeps and may attempt to flank you from the sides. Simply use Conflagration on him and Sear stun him, showing him who's King of the Lane.

Creeping:

Creeping is generally simple. In simple terms, the skill rotation to clear a creep wave is:
Pillar of Flame -> Conflagration

Pillar of Flame will allow you to apply your passive on a group of minions. Reading your Conflagration skill, it will spread to nearby creeps if cast on a target inflicted with your passive, Blaze. With sufficient AP, this will clear a creep wave in one combo.

Start off the game with a Meki Pendant and 2 Health Potion. This will ensure you have a relatively healthy supply of mana regeneration for your harassing skills and some health replenishing should you get hurt.

Turn that Meki Pendant into a Tear of the Goddess when you return to base and grab a Boots if you can. Turn those into a Ionian Boots of Lucidity.

If you are feeling confident in your ability to kill or at least land assists, I highly recommend Mejai's Soulstealer

Happily harass your lane while building your mana pool and get a Hextech Revolver Now Pyroclasm heals you everytime it bounces off an enemy and that passive 2% of enemy max HP in magic damage? Yeah, that heals you too.

Now work towards a Rylai's Crystal Scepter, starting with Giant's Belt to give you that long overdue HP survivability. Now all your skills slow, almost guaranteeing that no one can run away in team battles after Pyroclasm while you unleash hell upon them.

Finish off your Will of the Ancients whenever possible sometime around here.
--> Thanks to Political Squid for pointing out that the Health boost outweighs the minor spell vamp boost

Now, you just need a ****load of AP. What better way to do so than Rabadon's Deathcap?

Ok, you're almost there. Your Tear of the Goddess should be close to that +1000 mana max. Turn that sucker into Archangel's Staff and watch your AP soar.

Now go out there, and show everyone how a real hero lands triple kills like a pro.

Alternative Items:

Abyssal Mask - If you need that flat MR penetration for teams with < 100 MR.

Void Staff - If you need that extra kick in your spells in terms of MR penetration for teams with > 100 MR

Zhonya's Hourglass - If you find yourself targeted alot in team battles. It will buy you time for your skills to cooldown, and time for your teammates to damage them enough to get them off your back. Also good AP boost.

Rod of Ages - I've seen many Brands get this. Good source of HP and AP over time.

Deathfire Grasp - Grab this if you want to make that 4000 HP Cho'gath run in fear of you, or for that extra 200+ damage "nuke" skill.

Frozen Heart - CDR item. This will make you alot harder to kill with the +99 Armor.

Haunting Guise - Not really my item-of-choice but useful if you wanna bring your enemies MR to 0.

Lichbane - If you wanna push towers, or deal massive damage using basic attacks after casting a spell.

Morello's Evil Tome - CDR item + some good AP.

Situational Items:

Hexdrinker - I never find myself getting this, but if you like that 300 damage shield before dying, then go for it. I see people getting it just for Karthus

Quicksilver Sash - To get rid of pretty much every debuff, including Mordekaiser's Children of the Grave and Ignite. Great if you're being CC'd down.

Banshee's Veil - Good for messing up enemy skill combos like Veigar's Event Horizon -> Dark Matter -> Baleful Strike -> Primordial Burst. Also good MR source

Soul Shroud - CDR item. Not that great on you since you don't get much from it aside from CDR unlike the others.

9 x Greater Mark of Magic Penetration Magic Penetration for that extra edge in your spells.
9 x Greater Seal of Replenishment flat Mana regeneration to help you last longer in lane during Laning Phase.
9 x Greater Glyph of Cooldown Reduction flat CDR for those much needed, well, CDR early.
3 x Greater Quintessence of Scaling Cooldown Reduction CDR per level for that late-game CDR.

[+]Excellent farmer - ( Pillar of Flame-> Conflagration generally wipes out a creep wave)
[+]Excellent nuker
[+]Can stay at the back during team battles while raining hell on enemies
[+]Good at disrupting enemy rhythm during team battles while Pyroclasm wrecks havoc
[+]Great ganker if you can land his Conflagration + Sear combo
[+]Relatively good at escaping - (While being chased, Pillar of Flame behind you and Sear to stun the one nearest to you. May also Pyroclasm if more than 1 is chasing you to disperse them)

[-]Runs out of mana relatively easily without Tear of the Goddess
[-]2 important skill shots. Must be relatively skilled to bring out full potential.
[-]Relatively squishy, but can heal very well with Will of the Ancients

My Preference:

Ghost - For that I-need-to-get-the-hell-out-now button. Also great for kiting those pesky fools that think they can take the heat.
Ignite - Total synergy with your passive.

Other Viable Options:

Exhaust - I'd only suggest getting this if you intend on putting that point into Cripple under Masteries for that Magic and Armor Resistance debuff. The slow effect can be achieved once you acquire Rylai's Crystal Scepter where even your passive of 4s slows.

Cleanse - Always good against a team of heavy CC/disables.

Teleport - I will recommend this purely for ganking/pushing purposes. Wards and Teemo's shrooms make ganks ridiculously awesome, especially if your entire team has it. I swear people underestimate the versatility of teleporting to wards/shrooms in the bushes.

Flash - More of an escape skill, rather than an offensive skill. The idea is for you to blow AoEs in everyone's face WITHOUT incurring damage. Flashing into close range team battles will mess you up really bad.

Bad/Not Recommended Skills:

Revive - Unless you want/can backdoor like Twisted Fate or Pantheon.

Heal - Let a support get it.

Clairvoyance - Let a support get it.

Clarity - Let a support get it.

Rally - People still get this? I suppose Jarvan IV could, but...

Smite - Not really much of a jungler.

Fortify - Let a support get it.

Following this guide, you could potentially achieve the following:
[+]Max CDR (40%)
[+]About 900 AP (With fully stacked Mejai's Soulstealer
[+]About 25% spell vamp (You can heal off creeps easily with your arsenal of AoE skills and passive)
[+]Decent amount of survivability (~2000+ HP thanks to Rylai's Crystal Scepter
[+]Quad and pentakills
[+]Make an enemy or two ragequit.

Akali:
Common skill rotations they would use is Mark of the Assassin -> Shadow Dance -> Basic Attack -> Twilight Shroud -> Crescent Slash...etc

To counter, you'll have to be very precise. The range of her Mark of the Assassin is about the same range as your Conflagration. So try to cast it on her right when she casts hers. Then, if she follows up with Shadow Dance, since it makes a beeline towards you, a Sear in that general direction would stun her right after she hits you. Follow up immediately with Pillar of Flame and that should knock out a major portion of her HP. At this stage, she will either Twilight Shroud and run, or keep attacking you in hopes of out-DPSing you. If that's the case, repeat the aforementioned skill rotation and she will either run or die.

NOTE: Casting Ignite after that combo scares the **** out of most players into running away.

Alistar: