Irelia is a very diverse champion. She can serve as both the tank and the anti-carry for the team, and so her items must balance tankiness and damage efficiently. In addition, her laning phase is extremely powerful; when facing an overpowering opponent, she can tower hug effectively and make a comeback, and when dominating the lane, she can easily keep the enemy down and continue to faceroll.

This guide will explain different ways to build and play Irelia. Her items and item order can vary every game, so please read each explanation for each item route!

RUNES

The runes are pretty simple.

Marks: I find flat Attack Damage marks to be much more useful than Armor Penetration (ArmPen) simply because it helps to last hit a lot more. At a certain point, you will be unable to last hit minions after the standard 2 hits from tower (or 1 hit on the ranged minions) with ArmPen runes. Also, I've heard arguments that ArmPen is more effective since it makes Irelia's Bladesurge stronger. For this, I can say that Bladesurge has a very low base bonus damage, so the extra ArmPen isn't as effective as it could be. Also, I make it a point to not maximize Bladesurge as the first skill.

Seals: Armor seals are simply irreplaceable. Long ago, I used to use Dodge runes, but since that isn't an option anymore, I use Armor. This helps with early game tankiness, and I don't really see any viable alternative.

Glyphs: Magic Resistance blues are the way to go, as many solo tops still do magic damage. Also, other alternatives such as Attack Speed, etc. aren't nearly as useful. (what would you do against a top Vlad/ Kennen/ Rumble without MR?)
If you are extremely rich and are in a Draft Mode game against someone who deals only physical damage (Tryndamere, Garen, Darius, etc.) then I guess you could have Armor blues too, but for a standard rune page, MR blues are the best.

Quintessences: Again, flat Attack Damage helps to last hit a lot. I guess you could replace these quints with ArmPen, too, but flat AD is just my preference.
**Also, I find movement speed quints to be pretty useful. I've had many moments when I survived a gank purely because of the tiny bit of extra movement speed, and it really helps when kiting as a melee champion against another melee offtank solo top. Is very useful with your Q (Bladesurge) ability, since Bladesurge brings you into the minion wave and close to the enemy. With the extra movement speed, you can usually back out before the enemy's attack animation hits you.

MASTERIES

Irelia is not, or should not, be the sole "tank" of the team. Or if she is the only tank in the team, she cannot be expected to be as tanky as, for example, a Rammus. Also, she is expected to deal more damage than a Rammus, which is why I go with 21-9-0 Masteries.

First off, while the Movement Speed mastery is very tempting on Irelia, I don't find it as useful as starting with +40 HP and some extra MR/Armor.

Second, a 9-21-0 Mastery page doesn't seem as useful as a 21-9-0 page because Irelia is, by both Riot's judgment and mine, an assassin. Yes, a tanky assassin, but if I'm given the choice of having the power to kill or the power to live, I'd like the power to kill, since killing the enemy first will ultimately result in you surviving. This logic might not be the best, but I find it very suitable for the justification of my Mastery page. Basically, the 21 defense masteries are just not up to par with the offensive ones.

This is where things vary, so I'll make it as simple as I can.

Item Order

1. Almost always begin with Boots and three (or four) pots.
2. The next item is either a Phage or a Wit's End, depending on if the enemy deals physical v. magic damage. If you got Wit's End, buy Phage too, afterwards.
3. Finish the boots, either Ninja Tabi or Mercury's Treads, depending on your laning opponent AND the enemy team. Sometimes, though, I buy a Recurve Bow, finish the boots, then finish the Wit's End.
4. Always finish Trinity Force before getting a Warmog's Armor. Also, you can purchase your Atma's Impaler immediately after Phage/ Trinity Force. You do not have to get Warmog > Atma.
5. At this point, you have 1-2 open slots left (depending on if you got Wit's End). Having a Wit's End means that the enemy team deals a lot of magic damage, so I could get a Force of Nature, but by now the enemy AD carry should also be dealing heavy damage. Omen's or Force of Nature: your choice.
6. If you have Omen's, buy Force of Nature. If you have Force of Nature, buy Omen's. OR buy a Guardian Angel (I usually reject this item, however, as Irelia.)

Item Justification

Wit's End: you must remember that a lot of Irelia's laning potential and damage comes from her W (Hiten Style). Wit's End not only makes her lifesteal a lot more effective (as it is completely an on-hit passive) but it also makes her 75 true damage a lethal, or lethal-er, weapon. The extra damage from Wit's End is also very powerful, as many solo tops tend to build armor > mr, and building MR to counter a Wit's End on Irelia is just useless.
Atma's Impaler: Even after the 2% --> 1.5% nerf, Atma's is still a very nice item to have, especially when your maximum health will reach almost 4k. Also, the armor will help bring up some beefiness, especially if you went for Wit's End and Merc Treads > Ninja Tabi, because no matter how AP oriented the enemy team might be, they will still do some Attack Damage.
Early Atma's: A lot of people refute the awesomeness of an Atma's Impaler before a Warmog, but remember that with you will still have about 1800-2200 health with just a Phage or Trinity. That's still 30 extra damage, and the huge armor bonus is very effective early game. Also, the crit chance given is a nice bonus, since trinity already gives some.

Why no Gold/5 Items?? Because buying a Philosopher Stone/Heart of Gold will not help you lane any better. On your first return, buying a Phage or part of your Wit's End/lvl2 boots will help you farm and fight your opponent/survive a gank. Using 800-850, or 1650 gold, early game will lead to you feeding your lane or missing a lot of farm simply because you are weaker than the enemy. It simply isn't worth the investment. Scoring a single kill with a Phage beats the investment gold from a HoG or philo stone, because it gives you gold AND denies the enemy any for a short period of time.

Also, I find that a Wit's End and Phage makes Irelia a lot stronger than most champions who are stronger than her. Irelia is very item dependent, but a few items makes her exponentially stronger.

Here's how it is:

Max W (Hiten Style) first for sustain and fighting ability. The extra 30 damage from Bladesurge isn't worth losing on-hit lifesteal and true damage.

Max Q (Bladesurge) second for the decreased cooldown/level. At this time, you should be building your Trinity Force, so the extra damage from Bladesurge is technically doubled. The 2 second cdr/level and extra 30 (60, with Trinity) damage is much more useful than the 50 damage and 0.25 second slow/stun from Equilibrium Strike. This is mostly because I find myself using W > Q to harass a lot. Getting the Bladesurge to 6 seconds CD is an amazing improvement.

Max R: Self explanatory... it's your ultimate, and it gives decent damage, crazy sustain, and nice lane control. Get it.

Equilibrium Strike: I used to max this before Bladesurge, but Bladesurge is so much more useful for chasing. The slow from Strike is useless if the enemy just flashes or ghosts away, while Bladesurge allows you to stick to the enemy carries or whomever you are chasing.

Flash-Ignite

Ghost-Ignite

Flash- Exhaust

Ghost- Exhaust

...and that's it. The rest make you weaker in lane or aren't worth it. Best to be able to fulfill your role of killing the enemy carry.