This is the Lux I play. She is one of my main characters, and a decisive force in any team fight. Her combination of CC, AP damage and range are what make her amazing. I discovered the build on my own, played around with other people's builds from this website's guides, and eventually returned to this as my favourite.

The runes are pretty standard, although the big exception being that I switched the Quints for movement - personally, I feel that the extra MS is far more valuable than the minor bonuses in any of the other stats, as my Lux tends to be targeted. A lot.

The item build I show is the order I use in more or less all of my games, although I tend to vary between using another Archangel's Staff or Rylai's Crystal Scepter, depending on how my game is going. I usually go for the Mejais Soulstealer fairly early almost every game, as I'm quite the optimist! :P But also because Lux should rarely die when played right.

E ( Lucent Singularity) is her major lane control and harassing tool, and therefore my first skill. Use it frequently. Depending on the lane match-up, I use it to push creeps and farm, deny enemy champs their XP (by placing it near the caster creeps at the back, and forcing them backwards) and/or scout the brush. In bigger team fights, it's great for dealing damage to large numbers of enemies at once.

Her Q ( Light Binding) serves different roles in early and late game.
In the early game:

• it prevents enemy champs from dealing damage to you (especially melee champs)
• it can be used in combination with E to assure AP damage on enemy champs. Basically, when your enemy is near their creeps, throw an E down that places them roughly in the middle, when they start running, throw you Q so that it'll catch them if they keep running. That way, they either get hit by the E or the Q.
• and obviously, it's great for gank purposes to catch and hold a champ (or 2!) in place while your allies beat them up

In the late game:

• It turns 5v5 fights into 5v4 / 5v3 fights, if the right enemies are caught (for at least a brief duration)
• It saves allies from being chased down
• It deals tons of AP damage, so that even if you hit the wrong target, no biggy :P

Her W ( Prismatic Barrier) is useful, but not a game changer. It can sometimes mean the difference between an ally dying or escaping, and can be used to fool enemies into chasing after you (look at me, I have no health) and pulling them into a trap... but these occasions are rare, hence I level it last

Her ulti ( finales funkeln) is beautiful. She gets fantastic range on it (try moving the screen ahead of your character to see how far it goes, and to get accustomed to its range. It's roughly the distance from one tower to the next). It's obvious use is for finishing off low HP enemies, but this is far from it's only purpose. Especially with Ionian boots and blue buff, the CD at higher levels is so low (24s is lowest CD possible) that it's worth spamming for creep kills and enemy harassment. I've also stolen a few Baron buffs with it before, but this requires very good timing, assessment of enemy burst damage capability, and of course you have to know that they're at baron! I highly recommend you spam her ulti ordinarily (i.e. every time it cools down, you should have something nearby to hit), but keep it ready if you feel a team fight is imminent (unless you can harass them a little before the fight starts, and again during :D )

Flash is essential. It's very important that you are able to quickly, if not instantly, escape from damage. It can also be useful for getting in range to catch an enemy with a well aimed Q, but I don't recommend doing this unless you are sure of success and of not being ganked immediately afterwards ( Flash, Q, Kill, now I have no escape mechanism...)

TP ( Teleport) is good for it's usual merits of saving towers and allies, etc., but the main reason I love it is to TP into a smaller skirmish (2v2) and ulti any fleeing targets that your allies can no longer reach. It's also great for early game item superiority and mana regen - Once you have enough gold for your tear, and are running low on mana, kill the enemy creep wave, then B to base to buy your tear (and wards, potions, etc., as needed), and quickly TP back to your lane before any of their next wave of creeps has died. If you're 1v1 middle (which can be great for her, but depends entirely upon who you're laning there), you can keep pressing for mini-skirmishes with your enemy, until they're low enough for you to B home, heal, and TP back to kill them. TP basically ensures early game lane and item superiority.

Some important tips on playing Lux:

• Lux should have ridiculous amounts of AP by late game, but may sometimes feel more support-like early on, as this build doesn't always give her the highest early-game AP
• In team fights, stay in the back (as far away as you can manage, well still within range to throw your spells into the fight). Her auto-attack can be useful on enemies that you've hit with spells, but in big team fights it is not worth the small amount of damage you'll add to be in such a vulnerable position. She is a much bigger help alive and spelling than dead.
• Use her E ( Lucent Singularity) on creeps, especially caster creeps which can be hit once with E, and then killed with auto-attack, from a fairly early level
• If you know you're safe (to a reasonable degree anyways), don't feel afraid mid- to late-game to use your Q ( Light Binding) to farm. Its cooldown is reasonable.
• When you go back to base, throw spells while you're healing, as every extra bit of mana from your Tear of the Goddess will help you in the long run (although don't waste your ulti then :P )
• An enemy fleeing into the jungle can more often than not be hit with a well-placed ulti (their need to flee generally makes them less likely to zigzag through the jungle). Practice judging where enemies are going to run to (watch your allies when they're fleeing - their routes are most likely identical to those your enemies will use)
• W ( Prismatic Barrier)works well vs early-game Karthus! Throw it as soon as his ulti comes on, so that you get the double shield, rather than only a single)
• Wards are your friend. Lux is a fragile champ, and this build does very little for her defence. Ward ward ward ward ward ward ward ward ward. Ward!
• Get blue buff as often as possible. There are few other champs who will use it as well as you do (CD and mana = spell spam time!)
• Demaciaaaaaaaaaaaaa!

Pros:

• Great CC
• Fantastic range
• Farms very well
• Some support functionality

Cons:

• Fragile
• Relies somewhat on 1 or more capable physical damage teammates
• Tends to get targeted quite heavily
• She doesn't like Kassadin. No really, trust me.

I think I can predict some of the comments that might show up here. I'd like to address some of the preemptively...

Lich Bane can be a very useful item for Lux, but to be honest it just doesn't fit with how I play Lux. I don't like to stay within range of auto-attacks for any longer than needed (if I'm in range to hit with mine, many enemies will be in range to hit me with things I don't like). Lux does more than enough damage from afar, without the need to risk her getting too close. A distant Lux with 20 stacks does more damage than a nearby Linch Bane Lux with little or no stacks.

I think that glass cannon is a viable role. Yes, I'm sure that sometimes your enemy will outplay you and get around to killing you, but I think that if that happens too often, they are just better players than you, and probably would have won regardless of how tanky your Lux was (?!). I would also like to use this points to refer to a previous point... WARDS!

W ( Prismatic Barrier) could be leveled earlier, but the minimal protection it offers doesn't compare to the offensive capabilities of her other skills. The best defense is a good offense...

And lastly, "Demaciaaaa" is sexy.