Irelia's Song: http://www.youtube.com/watch?v=WEpMQldbPJM

Irelia is a great bursting Champion competing with Champions such as Akali, Pantheon or Xin Zhao. I build her to Jungle and ultimately be an immense threat to the enemy team when she goes out and ganks because of her incredible burst damage. And like the other burst champions, she is average in terms of squishiness and tanking. Here are some of the pros and conss she has:

Pros

+ Very great burst damage... can slay a squishy champion very fast
+ An assassin in taking out enemy carries with her burst damage and getting out quickly with her passive
+ Great with both Golem and Lizard buffs
+ Life steal very helpful in pushing and turning bad situations into good ones
+ If laning with a strong champion early game like Caitlyn or Garen, you can get First Blood easily with Equilibrium Strike
+ Fast ulti cooldown for what it's worth
+ Bladesurge allows her to solo lanes with some efficiency in last hitting minions

Cons

- No escape mechanisms out of Crowd Control except for her Passive of reduced stunned time.
- Team dependant in DPS
- Extremely mana hungry early game and sometimes mid game
- Most items she needs cost a fortune

Changes
Jan 30th, 2011: Added Champion theme song
Jan 19th, 2011: Changed Jungling route
Jan 19thm 2011: Exchanged Sheen with Madred's Bloodrazor in terms of what is purchased first

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I take Smite and Ghost as my summoner spells. Smite is a pretty obvious choice considering I will be jungling, and Ghost will mostly help me escape bad situations (ganks) and help me catch up to fleeing champions. You may think Flash should be used by all champions, well I would definitely strongly suggest you take Ghost over Flash with Irelia any day. Irelia's passive Ionian Fervor partially acts like a Flash; Henceforth making Ghost a better choice as it not only helps you run away faster, but also lets you catch up to champions running away, especially combined with Trinity Force. Flash can't exactly do that, as most of the time you'll flash in and the enemy champion will use one of their escape mechanisms.

I take a 21-0-9 Mastery Tree taking all the crit masteries. I ignore the Archaic Knowledge mastery as all of Irelia's abilities except for her Ulti use Magic Damage. And I usually only use my Ulti on a farming wave prior to just completing a gank. The 9 in Utility is also standard jungling masteries as well.

I take Hiten Style as my starting skill as it will be your most efficient ability to use in terms of jungling. Then I level my Equilibriam Strike obviously to deal more damage. I proceed to up these 2 abilities to Level 2 until I get my Bladesurge at Level 5 which will enhance my ganking effectiveness. From then on I level my Ulti whenever I can.

I take Attack Speed Marks, Armor Seals, Cooldown Reduction per level Glyphs and Attack Speed Quintessesences as my Runes. The Attack Speed will give me that much more burst damage while the Armor is obvious for jungling and the Cooldown Reduction per level will quicken my waiting time for my Ulti which is already on a short cooldown late game.

Early Game:

I take the classic Cloth Armor and x5 Health Potions as my standard jungling starting items. My starting jungling route goes like this: Wraith Camp>Wolf Camp>Golem Buff>Golem Camp>Base>Wolf Camp>Lizard Buff>Wraith Camp>Golem Camp etc. etc. I buy Madred's Razor to increase my effectiveness in Jungling and taking Dragon at level 10ish. My next item would be Berseker's Greaves to enhance my burst damage. Now I know for a fact that it is critical to buy Sheen with Irelia as it gives her burst damage/ganking effectiveness that much more power. I used to think that I should gank using a Q/W/E combo (similar to my primary champion Xin Zhao)... of course I regret doing so. Combined with Sheen, you should use a combo of W/Q then use E wisely. Using your Hiten Style on the enemy champion will activate your Sheen increasing your base damage by 100% for your next attack. Your Bladesurge goes like this: Irelia dashes forward to strike her target for 20 / 55 / 90 / 125 / 160 (+1 per Attack Damage). My Sheen combined with that +1 Attack Damage will increase my Bladesurge's Attack Damage greatly! Like I said, you should use your Equilibrium Strike wisely... if you have seen your enemy champion blow most of their abilities already before you ganked them then it's best to just use it after a 2 second delay so you can activate your Sheen again, and get a 100% plus damage auto attack on them. After your Equilibrium Strike you will probably have your Bladesurge back from cooldown. This initial burst will get my ordinary target to around half health in which case I'll hit my Equilibrium Strike again since it's at a 9 second cooldown, where immediately after I'll Bladesurge them to use my Sheen effectively. I'll then use Hiten Style and or Transcendant Blades to finish them off. But also remember that your Sheen gets activated for each Spirit Blade you fling at them.

Tip: Jungling obviously pays it's toll on you in terms of health. This isn't a bad thing when trying to setup a gank, as you can Bladesurge in and get your Equilibrium Strike off quickly to stun them for you and your teammates to finish off.

Mid Game:

I finish my Madred's Bloodrazor which gives me a bit more Attack Damage and a load of Attack Speed combined with the passive which kills Golem Buff, Lizard Buff, Dragon and enemy tank champions a whole lot faster. I then buy a Phage to reduce the number of times I see champions run away from me and Zeal to ultimately buy my Trinity Force. With your Trinity Force you're definitely doing alot of damage output combined with that 12% movement speed included in it, which makes you that much more effective in terms of chasing and escaping which is one of her downfalls.

Tip: As soon as you got your Madred's Bloodrazor, you can harass tanks with alot of health with your burst damage and passive of the Bloodrazor.

End Game:

Your Trinity Force will sometimes be your end game, but in the cases that the game extends even more and you have 3 slots still yet to fill then it's best to buy a Guardian Angel. The reason behind that is because I basically have no Magic Resistance in my build at all, and by getting Guardian Angel I can help improve that, and also improve my Armor and revive myself if I do end up dieing. If the game extends even further, then I really begin to be a pain in the *** after buying 2 Starks Fervors to give me Lifesteal and basically wipe out their whole team in one swift push.

Tip: You can burst and slay the enemy carry whome is baiting his teammates and utilize your Guardian Angel to revive you and kill the rest of they're team