Many people consider Irelia a Hybrid carry. This may have been true before they switched Irelia's ultimate to physical damage instead of magic Damage.

PROs
Great amount of damage
Stun when low health (lower than enemy)
Amazing Lane sustainability
Decently Tanky late game

CONs
Low damage output early game
Mana issues "early" game.
Squishy early game

All in all, Irelia should be played as a farming champion (yes that may get boring) early game and an assassin/off-tank late game. She should always be played in the solo lane top.

I go armor penetration to get the full benefit out of my auto-attacks, bladesurge and ultimate.

I prefer straight armor/magic resist runes for the extra defense early game and late game
And of course armor penetration quints.

You may like to use Attack damage per level (or straight attack damage) but I find armor pen to increase my damage output more since her spells aren't the best of scaling off AD.

Other options include:
Seals: Mp5 (useless late game)
Attack Damage
Or w/e else works for you.
Glyphs: Cool-down reduction. Useful since you don't have any from masteries/items although blue could easily solve this.
Ability Power (if you prefer hybrid Irelia)
Quints: I highly recommend Armor penetration quints as these are the best option but.... Health quints for survivability

I prefer 9/21 getting armor penetration and then moving down the the last one in defense. You could go 21 in offense but the survivability that defense masteries provide allows you to stay in a fight longer and do more overall damage.

The part that every probably came here for xD

CORE:

Mercury's Treads Great CC reduction allow you to continue in fights with little trouble. Ninja Tabi is also a good choice if they are AD heavy.
Trinity Force This is probably the best item for Irelia. Bonus attack damage after using a skill, movement speed, health, mana, attack damage etc. What's not to love.
Bloodthirster This is great because it adds tons of AD to your skills/auto-attacks and also allows you to heal even more with lifesteal. I recommend farming this everytime you die to get it to its full potential.
A substitute for this item would be Zeke's Harbinger, give great lifesteal and attack speed.

Situational Items

Now after this you have to consider the enemy team. Do they have a AD carry that's just dominating you? then i would suggest Thornmail

Are you having trouble with CC even after your passive and your merc treads?
Banshee's Veil

If the teams are balanced then I would probably recommend Banshee's Veil and Randuin's Omen, which is almost the core build, but as these items can vary, they did not make it into the core build

Sixth Item

In the build I have Infinity Edge as the last item because this is what I almost always get as my last item.

If they have an entire team stacking Armor then Last Whisper is the way to go.

Zeke's Harbinger Would also be a great choice if you feel that a little more life steal would make you invincible

And finally Guardian Angel I recommend this item in situations where you can either: A.) Rush in and easily kill the carry(carries) within a couple of seconds or B.) You have a relatively high damage team and not having an infinity edge/Stark's Fervor wouldn't matter.

Items that may seem good but really aren't:

Hextech Gunblade
WHAT?!?! "But Irelia has skills that scale off both AP and AD. This would be the perfect item for her!"

The reason for not getting that is because only one of your skills actually does magic damage, and the scaling on AP isn't really that great. So spell vamp is almost completely useless and the AP also not really useful. Yes it would benefit Irelia but infinity edge would be way more useful

Guinsoo's Rageblade
This item isn't entirely bad. The AP would help "slightly" and the attack speed is nice. But overall there are other item choices that are much better.

Lichbane
Again, AP isn't that great on irelia making this item pretty useless. And Trinity force is way better.

As you can see I prefer Irelia with Exhaust and Flash just because both of these provide offensive and defensive cool downs that can save your @$$ or help get that kill.

Other good ones include:
Ghost Useful for catching an opponent or running away
Ignite Useful for finishing an opponent off
Cleanse Stunned? No problem
Surge Faster attacks and stronger E/Ult (ignite/exhaust would be better)
Heal Critical moments where you just need a little more health

Probably the most important part of the guide. Yes you can buy the best items, have the best runes and even have the best skill sequence yet still fail horribly.

Early Game
Early Game is slightly boring for Irelia if in the solo lane(which you should always try to take). You pretty much just sit back and farm Creeps, using Q if needed to avoid harassment and auto-attacking to restore health using your W's passive. If paired with a partner, still to farm, but save your mana for bladesurging and stunning.

Mid-game
This starts around the time you get sheen/merc treads. Now you are not invincible, but you deal significantly more damage because this is when you have your ult. Now you should be harassing more often, using your ult to heal if needed and trying to kill that pesky champion in top lane. If top is empty go gank! Irelia is great at ganking with an instant teleport and then a slow right on top of that. Pair that with exhaust and you my friend are ready to hunt some champions.

Late Game
Late Game starts around the 15-25 minute mark regardless of what you have. This is when a couple of towers are down and you are starting to fight as a group. You should usually try to hide in a bush and wait for your tank to initiate, but don't be afraid to initiate yourself if you see a squishy over extended. Irelia can take some damage if you pop your ult on as many champions as possible.

Solo-Fighting:

When 1v1ing a champion, try to harass if possible. Bladesurge, pop Hiten Style
and then use Equilibrium Strike and walk away. Don't be afraid to just keep auto-attacking you think you can win the fight.Don't forget about Equilibrium Strike and its stun. Sometimes it's better to save it for when they have more health, because this can easily turn around a fight. And ALWAYS save your ultimate to finish them off. Usually most champions forget about your ult and stay thinking they can kill you. Little do they know that you can heal 1/4-1/2 your health and deal a great amount of damage at the same time!

Team-Fights:
If being focus'd try stun whoever has more health, pop Bladesurge and auto-attack to heal. And don't forget about your ult! use it when they attack you to ensure you can get it off and heal properly.

If not being focus'd, Bladesurge their carry and just have fun killing them. Use your ult and Bladesurge To hunt down any1 who tries to get away.

Please leave any advice/questions in the comment section below! I'd be more than happy to revise my build if I find it a good suggestion, or to help make things more clear.