Needless to say that there are a lot of Irelia builds, but the majority (+-80%) build her like a standard off tank: a boatload of armor and magic resist with very little attack damage. These builds make Irelia hard to kill, but they don't allow her to dish out massive amounts of damage. That's what The Blade Weaver build is for. With this build Irelia is still bulky but also has amazing dps for team fights.

Pros:

+amazing dps
+great survivability with ls and self healing w and r
+cc barely does anything to her
+Amazing solo lane
+high ms for those runners
+nice cc for team fights and 1v1

Cons:

-focused like a boss
-dependent on last hits/kills
-item dependent
-melee
-squishy-ish

For team 1

Greater mark of desolation provides armor penetration, which is very effective on ad champions like Irelia. Armor penetration allows Bladesurge and Transcendent Blades to cut into an enemy's armor and deal more damage than they would otherwise.

Greater Seal of Armor provides flat armor that makes Irelia more bulky early game, and is especially helpful when in a solo lane.

Greater Glyph of Magic Resist adds flat magic resist which again increases Irelia's bulk, making her harder to kill with spells.

Like the greater mark of desolation, greater quintessence of desolation adds more armor pen, which will help Irelia cut through an enemies defense.

For team 2

Greater Seal of Mana gives Irelia more mana, which she definitely needs because her abilities drain her mana very quickly.

Greater Glyph of Mana increases Irelia's mana even more.

0/9/21 or 9/0/21 tree to utilize Irelia's abilities more, but I decided on a 0/21/9 mastery tree to give Irleia some bulk so I can buy offensive items. If you decide to use a different set of summoner spells, make sure you adjust your masteries appropriately.

Great on Irelia

Flash is always a solid choice for any champion. It allows Irelia to escape from a losing battle or jump into a weaker champion.

Ghost lets Irelia race towards a fleeing champion or escape from a gank with ease.

Exhaust hinders an enemy helpless against Irelia's barrage of attacks and cc.

Ignite is nice against champions like Tryndamere or Fiddlesticks that can life steal or drain for a lot of health.

Ok on Irelia

Heal, is viable on a soloing Irelia so that she can remain in lane longer.

Irelia is very mana dependent, so Clarity can be quite helpful.

Teleport allows Irelia to jump into a team fight at a crucial moment or protect a tower that is under siege.

Bad on Irelia

Irelia is neither a support or a tank, so Fortify can be replaced by much better spells.

Rally is easily destroyed, avoided, and generally stopped. Use it pretty much never.

Irelia's passive, Ionian Fervor, when coupled with mercury treads makes Cleanse a waste since cc hardly effects her anyway.

Clairvoyance is a spell for supports. Irelia is not a support.

Revive can only be used when you die. Dying is a bad thing that shouldn't happen often, if at all.

Smite is for junglers only, and Irelia is not a jungler.

Ionian Fervor is one of the best passive abilities in League of Legends. When 3 enemy champions are around cc on Irelia is reduced by 40%!

Transcendent Blades scales off of both ap and ad, so when Irelia's build is finished, each blade will deal367.5 physical damage to each champion it hits and Irelia will gain 91.875 health from each person it hits. That is quite a bit.

Equilibrium Strike may scale off of ap, but the cc it deals is great early game. at level 18 it deals 307.5 magical damage and has a 60% slow for 2 seconds if you have more health than your opponent, or stuns them for 2 seconds if you have less health. This ability is great for winning a losing battle.

Bladesurge is a dash attack that is perfect for last hitting minions and for catching up to champions. By level 18 it deals 440 physical damage to an enemy. It hurts and even restores mana and refreshes its cool down if it kills what it hits.

Hiten Style's passive is a sweet 26 health restore for each autoattack. Its active is even better: for 6 seconds Irelia's autoattacks deal 75 true damage.

Starting with a Regrowth Pendant lets Irelia stay in lane longer than without it, and a Health Potion ensures that she won't die easily.
philosopher's stone
Philosopher's stone and Heart of Gold combined give you an extra gold each second, and by the time you sell them they will have generated an extra 1800 gold. That's a lot of extra gold.

Mercury's Treads, when coupled with Ionian Fervor, makes cc against Irelia almost non existent since cc is reduced by 61%.

Trinity Force is recommended for a reason. It gives amazing stats and a great passive that reduces enemy movement speed.

Bloodthirster gives 100 ad and 25% life steal when it is stacked. With that good of stats, it's a must in a dps Irelia arsenal.

Force of Nature adds an extra 76 magic resist 40 health regen per 5 seconds, movement speed, and a great passive. What's not to love when you're facing those pesky ap nukes?

take your pick: Frozen Heart has 99 armor, 500 mana, 20% cooldown reduction, and a passive that reduces enemies' movement speed. Randuin's Omen has 75 armor, 350 health, 25 health regen per 5 seconds, 5% cooldown reduction, and a great active for team fights. You can't go wrong with either one.

Infinity Edge gives you an extra 80 ad, 25 critical strike chance, and 250% critical strike damage. It's the topping on the cake that is total annihilation.

After I played as Irelia using the top builds I realized something: Irelia is meant to take out the opposing team, not soak up their damage as an off-tank. That's when I created The Blade Weaver. I hope you try it out and give me feedback on what you thought because I think it works really well, but I am slightly biased.