Her passive gives her level-scaling on-hit magic damage (attack speed/bruiser/AD builds), her Q gives her AD scaling damage and sustain that procs on-hits (AD/bruiser builds), her W gives her high AP and high AD scaling damage and damage resistance that can go up to 100% with AP scaling (AD/AP builds), with an E that has a stun and high AP scaling AOE damage (AP build), then an ultimate that does high AP scaling damage and can disarm the enemy with a wall that does more AP scaling damage if enemies run into it (AP Build). With her W,E,R she can viably build full burst AP and still have a bit of other stuff from her other kit that somewhat helps (minor sustain/waveclear, mobility, and a passive that gives you a bit of auto-attack damage to help you duel/secure your bursted kill). She can also go viable AD (not sure if she can go burst AD, I don't think so based on her AD scalings), on-hit attack speed, and bruiser builds also. All of them benefit from some part of her kit, which the parts that don't directly benefit her build still give her what she needs (such as mobility, a stun, a disarm wall, dmg reduction, etc.).
Irelia really has the most balanced (as in build-path) kit I've ever seen in any game.