Alright, well, I've been playing LoL for quite a while now, and though I'm not a pro player, I think I know a thing or two about this game. I currently main Irelia, as she is my favorite character. She fits into many roles for the team and can really change the tides of a battle.
I included 3 different ways of playing Irelia here, if you don't like one of them, please dont downvote :P, it is my 1st build.

1st build - offtank/very tanky dps
2nd build - assassin/anti-carry(if done right)
3rd build - jungler - (*note: you can build it into any of the 2 above. I like offtanking better, so I decided to put the build as the offtank one.

Pros:
-Great solo top or bot
-amazing damage all game
-deals massive amounts of damage while being extremely tanky (offtank build)
-often turns the tides in a teamfight
-awesome ganking abilities -+

Cons:
-Her jungle isn't the fastest - you need a leash
That's all I can think of. IMO, Irelia is great in general
if you can think of something, PLZ comment and let me know :D

I like Ghost and Flash, because with theses spells, I can chase down almost anyone.
However, It is fine to use Exhaust or ignite to subsitute for ghost or flash, but I don't think Exhaust and Ignite is a good combo.

So basically:
Ghost + Flash
Exhaust + Flash
Ignite + Flash
Ghost + Exhaust
Ghost + Ignite

I would suggest you stay away from the other spells. They have their uses but I think that these 4 spells are the best on Irelia.

Offtank - 9/0/21
get increased buff duration. buffs are great on you. I do get magic pen, because Irelia does do a bit of magic damage, so I think the 15% is worth it.
Of course, you can go 0/21/9 or 9/21/0. I think it all works well. Personal Judgement will be the case for this.

EDIT: I changed it, as I do think 15-0-15 works quite well.

Assassin - 21/0/9
Pretty obvious here. You want lots of damage, and you get buff duration. I don't think you should take anything else here.

Jungler - 0/21/9
For solo queue, I usually take this, because as I said, Irelia's jungle isn't the best in the game, since you do need some help taking down blue. Get a leash, and with the extra tankiness from the masteries, you should be fine. Irelia's jungle gets a lot faster once you finish blue.

When I play arranged with a group of friends, I get them to hit it a few extra times, maybe take off 1/5 of blue's hp before I tell them to leave. If I do it this way, I can go for an 21/0/9 mastery page, getting more damage, Mpen and ArPen for later on.

You have lots of choices for runes, but for marks I think it is pretty simple
ARMOR PEN!!!!
Before the Irelia patch, you could play her as a hybrid ap/as anti-carry, but now, she's just a straight (tanky) dps.
Duff suggested that she can still be built as AP. You can, but that's not what this guide is about, and I personally find tanky DPS more enjoyable

For seals, I usually take Mana-regen when I'm laneing and armor for jungling.
The mana regen lets me harrass well with the bladesurge + equilibrium strike combo and allows me to spam bladesurge to last hit
Armor runes for jungling is obvious. Attack speed could work as well, but I prefere armor, since I get attack speed glyphs instead.

Again, for laneing, I take Mres glyphs. I think that this is the best choice. I don't find any other glyphs very useful.
For jungling, I take attack speed. You'll need it. Seriously.

I think that I made the items pretty clear, so the only thing that might differ is force of nature. If they enemy team consists of nukers and people who rely on hitting combos (ie: brand) banshees is a good option. I just love FoN because it give you lots of Mres and movement speed.
With this build, you should have around 480 movement speed. If the enemy team is getting carried by lots of DPS or AD, then consider getting a thornmail instead of the FoN. But if it is a balanced team, then I think my build should do well.

If you find that the enemy team is stacking health (like noobs) or they just have incredible amount of health and its annoying the hell outta you, get a madreds bloodrazors instead of atmas. It should help.

"But Jak, you don't even get wiggles for jungling!!!"
Yeah, well, its a good item, but I find that I just end up selling it. I keep the razors and build it into a bloodrazor, because late game it will definately help you a lot more.

You may have noticed I rush a sheen for all my builds.
The answers is: Sheen is great
Sheen is amazing on Irelia.
The proc effect will activate during ur bladesurge, allowing you to burst like a *****. the mana basically solves all ur MP issues. Build it into a trinity. it is Irelia's core item.

I changed this up completely, as purpleges suggested. I tried to max hiten style second, and it worked great. This is ideal on every build, as it gives you so much more damage output and makes you a great fighter. If you're solo laneing, the passive on hiten style is great for healing up. Equilibrium strike just doesn't do enough IMO, so I just take it at level 1 to avoid sticky situations. It may get you first blood, or it may prevent the enemy from getting it. After that, I don't level it up until later on.

Here is how you should max ur skills:->->->

Farm Farm Farm Farm Farm

Bladesurge minions if you can't get them. Take as many last hits as possible. Bladesurge allows easy last hitting. As long as you don't spam it like Karthus's Q, and actually get the mana back from the passive, philo stone and mana regen glyphs should last you the lane.

Duo lane should be easy for Irelia, Harrass with the Q+E combo, run back.
If you wanna go for a kill, Q+E+W. This is situational, because if you think that you can autoattack many times before they run out of range or die, then use it after E.
However, if you want to burst harder with Q and E, and perhaps there isn't enough time to autoattack, then please, use W+Q+E. Ult if needed. Don't be afraid to use your ult. It only has a 60 sec cooldown. Use it to farm a big wave of minions or just to heal up.

If ur solo laneing, just last hit with q. Don't go too far with harrassing. If they pushed you too hard, use your ult to clear the wave and regain lots of health.

When you're offtanking, wait for your tank to initiate, if you are the tankist hero on your team, then you can initate. Try to protect your carry, stun and burst down enemy carry.

When your the assassin or the anti-carry, wait for your team to initiate. Try to not take any damage before you go in the fight. After the teamfight starts, run though their team (with 75% CC reduction, it should be easy) and bladesurge onto that squishy carry at the back. (ie: ashe, cait, ez) and burst them down. If you have a competent team, you should win the teamfight 90% of the time. If your team lets you down, well... That's not my fault.

If you were jungling, it depends how you build. If you built into offtank, read the 1st one.
If you built into assassin, then read the 2nd.

There are a few ways to jungle with Irelia, but i prefere to start at blue. (remember to drink potions when nessessary.)
You do need a leash, but that's no big deal unless you're in super high elo or something.
After you finish Blue, Go to wolves. Spam skills, "E" the big wolf.
Go to wraiths. "E" the big wraith
Go to double golems, kill those (you might need to smite one of them)
At lvl 4, with all 3 skills ready, go gank.
Simply wait until your teammate is ready, Q, E (hopefully it stuns, if not, it will slow) and W and just auto attack. If you're teammate is competent, then you should get a kill, or at least burn the opponent's spells.

This is it. I know I have a lot of text here, so thank you for reading (if you did XD)
I hope you learned a bit more about Irelia and please, PLEASE try the build first before downvoting it. It is my first build T_T

Anyways, If you have any questions or something you feel like I should add, please comment or send me a message~