UPDATED: 3/16/2011 concerning Irelia nerfs.

Irelia is a very versatile champion and can be built differently to adapt to varying needs in any game. Irelia's main function in LoL is to 1. DPS 2. CC. A third option for Irelia might be to tower or to jungle. But these aren't Irelia's forte necessarily since other champions can fill that role better than she does.

A skilled Irelia player is the most noticeable early-mid game. Late game, Irelia's output and survivability depends much less on skill. Simply put, I have seen newer Irelia's on ranked games feeding early game but still manage to pull out decent end-game scores.

There is no set way to build Irelia but Irelia is the strongest with a 21/0/9 mastery and with core items including Trinity Force and Mercury's Treads.

This guide will attempt to break down different scenarios and help you analyze how you should build Irelia. Hopefully, it will help you achieve consistent results with this champion, not just "oh! i went 26/4/11 that last game. I own with irelia! GG"
Solid results can be just 6/0/2 or 9/2/5. This is especially true if you end up with a team which on paper loses on champion picks.

Starting items

With each specific set of starting items comes with a playstyle appropriate to it. For instance, with offensive startup items, you need to play offensively or you will lose the lane. Defensive items allow you to become resilient to harass and when you reach a certain level you can begin playing aggressively.

1. Doran's Shield

Doran's Shield is possibly the best item to begin with. For 475g, you get 120 HP which is useful to even keep till late game. The armor is useful since most harassing early game comes from physical auto-attacks. The regen is nice as well. If you don't know what you are facing, pick this.

2. Long Sword + Health Potion

Alternatively, you could get Long Sword + Health Potion, this gives you extra harassing power, providing that you pick Bladesurge and level it early. This serves great dual purpose - improves last-hitting minions and early killing. The health range of being able to refund Bladesurge cost on low-life minions will increase if you pick this item. You can improve your ability to surge to a minion and then a champion. Lastly, it builds into Phage, or Madred's Razors.

3. Sapphire Crystal + 2 x Health Potion

A third option is to build Sapphire Crystal first and lead that into Sheen. I would only do this when 1. your opponents have no CC and 2. Weak harassing abilities. I am not sure how often this might be - maybe as often as there are 2 afkers on the other side I guess. The extra mana is not necessary - you shouldn't need to last-hit with Bladesurge that often otherwise you'd risk over-extending. Harassing a lot against other champions poses you vulnerable to ganks which Sapphire Crystal provides no defense against. I've seen this option selected vs. Cho'Gath who also picks the same startup option.

4. Cloth Armor + 5 x Health Potion

A defensive setup. Good against heavy auto-attackers. Can build into Madred's Razors or Randuin's Omen or Frozen Heart.

5. Regrowth Pendant + Health Potion

This provides awesome health regen. It's good vs. burst casters. Builds into the ever so popular Philosopher's Stone or Force of Nature.

If you need to play defensively, pick Doran's Shield which is most often the safer bet. Not dying and staying in lane is the key in the laning phase.

Mid-game items

To play Irelia effectively, you MUST build DPS. While building survivability is viable with Irelia, you are the most effective in dealing damage and CC. Irelia is expected to kill squishies within 2-3 seconds since most CC's last around that duration. Building tank straight off will allow you to stay around for longer but that is often unnecessary. Irelia's survivability comes from CC reduction and Transcendent Blades and the ability to deliver CC in return.

Boots

Mercury's Treads - most effective. CC reduction means more mobility, more output, and more survivability all-in-one. If you are in a lane with a lot of enemy CC, this is a must. Some people suggest building Sheen before this but it all depends on what your opponent is building at the same time. If your opponent has upgraded boots, flash, ghost, or other speed boosts, getting Sheen will help the initial burst but will falter if they Exhaust or slow you. In ganking situations, the boots will allow for a multi-kill more easily than Sheen. EDIT: Irelia's base movement speed is no longer the best. It's 320 at start, consider this when you think about not upgrading boots.

Berserker's Greaves - Attack speed is great on Irelia but only later game. This is true if you intend to level Bladesurge up first. If you are facing tougher opponents with a lot of armor, it is actually smart to level Hiten Style first. The benefits of attack speed for Irelia is not strong early game. If the other team has very few CC's then you can consider this.

Ninja Tabi - this is a pair of cheaper boots. Get this item only if you are facing a lot of AD champions - your opponents consists of about 3 AD auto-attack heroes.

Sheen and Phage ordering

It's important to build Sheen as early as possible. This is because of the Sheen effect which allows you a critical attack as often as 2 seconds. I'm not 100% sure, but I think Bladesurge consumes one charge of this effect. This can allow you do a massive charge damage if either Equilibrium Strike or Hiten Style was used first.

EDIT: I tested this - Bladesurge will not do more damamge if W or E was used first. This is because Bladesurge creates and also consumes a stack Sheen effect.

EDIT: due to recent AP ratio nerf to Equilibrium Strike, Sheen would provide much less burst early on.

When you have both Mercury's Treads and Sheen, you can properly execute a gank or kill - you simply have enough damage. If you chose Phage first, keep in mind that the slow effect is by chance and not guaranteed. Having Phage first, you can leave your Equilibrium Strike out of the burst combo and you can chose to use that ability on a support champion to prevent him/her from helping your target. Because without Phage, more often than not, you would need to use your Equilibrium Strike as soon as you surge in. In most cases, Sheen will only proc only once or at most twice per gank.

After Trinity Force

Trinity Force provides the staple of your damage throughout the entire game. It is scaleable with almost any kind of item you are going to pick up later. Depending on the game, there may be times when you NEED to stop completing Trinity Force right after Sheen or Phage to get situational items such as Warden's Mail or Madred's Bloodrazor.

EDIT: due to recent nerfs to both HP and movement speed, Irelia will need more defensive items than before. The burst DPS loss from Equilibrium Strike may require that you need to stay in longer in a fight than usual. However, the -1 second CD reduction on Equilibrium Strike may prove useful in providing your team an edge in CC.

Going offensive

The game is at least balanced or even in your favor. This is when you should start dishing out unstoppable damage. Consider the following items:

Phantom Dancer - Get Zeal first and then see where the game is heading. If the extra movement and attack speed is working well, complete it. This item scales incredibly well with your Hiten Style which is very effective against tougher targets. Pick this if you are against 2-3 tankish champions such as Trundle.

Youmuu's Ghostblade - The attack speed provided on demand does not match that of Phantom Dancer. ArP is not that useful at this point of the game since Hiten Style far surpasses it in its role. Overall Irelia is a more consistent DPS champion relative to let's say Master Yi. Yi could use this way more effectively than Irelia. Get Avarice Blade first if you think the game can last at least 20 more minutes.
EDIT: this item may in fact be quite useful now since Transcendent Blades deal physical damage. I would suggest getting The Brutalizer first.

Infinity Edge - A little too expensive. This item provides a lot of boost to Bladesurge but its crit effect does not stack multiplicatively with the Sheen effect.

Bloodthirster - Again, Vampiric Scepter is weak compared with the Hiten Style passive. It would be nice when you can get those bonus stacks but then again if you are winning by that margin, picking almost any other alternative is just as effective. I honestly think Attack Damage scales worse than Attack Speed later game especially because Hiten Style gains both armor-ignoring damage and lifesteal from attack speed and not attack damage.

Black Cleaver - This is a good item. I would prioritize this right about the same level as Phantom Dancer. If you are facing weakly armored carries who have invested in some form of armor, this is your best bet around that. The great thing about this item is that it can create negative armor on opponents which makes you deal bonus physical damage. So this is useful even if your opponent is not stacking armor. However, this item is not strong against tanks.

Last Whisper - Almost an unnecessary item for Irelia. Too few opponents except against tanks draw a meaningful contribution from this item. You deal more damamge through the Hiten Style active.

Madred's Bloodrazor - Effective against HP stacking targets. The attack speed is nice and scales well. 3800g is almost too expensive for practicality reasons. But if you think you are losing to tanks such as Cho'Gath or if no other teammate can effectively carry this, you might as well starting building this as soon as you complete Sheen and/or Phage if you see any opposing carry rushing Warmog's Armor.

Zeke's Harbinger - nice cheap item which provides attack speed. Again the lifesteal isn't very strong for Irelia. Get this if your team is rather AD heavy.

Situational but cheap items for the mid-game:

Executioner's Calling - Opt for this item if you are against 2+ healing champions. Soraka, Dr. Mundo, Sona, Olaf.

Sword of the Divine - Get this if Jax is doing well. It's also strong vs. tanks early-mid game. Lackluster late game.

Going defensive

When you are not sure if you can really win the game or if a certain enemy champion is clearly fed, you need to adapt with items to turn it around. With Irelia, this flexibility is possible.

Randuin's Omen - If your main job is to pin down Ashe or any ranged champion, get Warden's Mail and see if it is effective. In my experience, Phantom Dancer simply does not provide enough survivability when pursuing ranged enemy champions which need to die ASAP. This itemization also counters Tryndamere, Master Yi, Jax for example.

Banshee's Veil - Possibly one of the best anti-magic items in-game. Useful vs. CC or any single-target nuke which needs to be negated such as Requiem.

Force of Nature - Build into this item if you had to take Negatron Cloak early on. It provides a neat movement bonus as well. The HP regen is lackluster.

Guardian Angel - This item could be GOD or meh. It gives you good defense stats and is pretty cheap. It's generally not very useful until very late game when death timers are so long. But what it really does is it stops your opponents from targeting you first. Therefore you should only get this if your DPS threat is so high that they need to focus down you no matter what. You really only need this to protect your killing spree for the gold.

Spirit Visage - I have seen Irelia's with this and Wriggle's Lantern and tower like crazy. They are not good teamfight items but unless you are planning to lane all game and push towers it's not viable.

Atma's Impaler - This thing is only cost effective in terms of damage if you have over 2600 HP. If you went Trinity Force, you probably still need another HP item before you get this. I only see this item useful if you are mostly up against AD champions. Or if you had Sunfire Aegis or Frozen Mallet.

Going completely tank

It is possible to build Tankrelia. But you'd be doing like no damage until you get Atma's Impaler. The sequence would be (after boots) Sunfire Aegis --> Randuin's Omen/ Force of Nature --> Phage --> Atma's Impaler --> Frozen Mallet. It is quite effective to alleviate some pressure off your main tank and protect some squishies without being too vulnerable at the same time. But this build is not very strong vs. casters. If you build this way, you are mostly harassing and trying to CC opponents or take some CC yourself.

REVISION:

After boots, get Sheen --> Phage --> 1. Negatron Cloak if you need MR or 2. Chain Vest --> Frozen Mallet --> Banshee's Veil or SELL Sheen to get Atma's Impaler. It does seem that Force of Nature isn't as attractive since it provides no HP boost for your Atma's to take advantage of. Randuin's Omen is still nice if you up against ranged AD carries.

The reason to sell Sheen is because at level 18 it provides about ~120 attack damage per 2 seconds under ideal conditions while with Frozen Mallet, you'd have ~2900 HP which equates to about 58 AD @ ~1 attacks per second if you get Atma's Impaler.