When you're in game, you can hover over your abilities to see what they scale off of. Orange numbers indicate attack damage while green numbers indicate ability power. You would see that her Q and Ult scale with AD. So, these abilities' damage is increased with the more AD you have. Her E and Ult scale with AP, too. But, since her W is an on-hit effect, and her Q applying on-hit effects, the vast majority of Irelia players go AD.
There is really only one item that you will rush every game, Trinity Force. All the base stats benefit Irelia, as well as the abilities on it. The bonus damage after casting an ability is an on-hit, so it applies to your Q. This makes CSing WAY easier, as well as gives decent burst damage for quick trades in lane. After that, it's basically all situational. Here are some options:

(Generally pick 1-2 Damage and the rest tank, because you'll deal enough damage that you don't need more)
And don't forget boots. Ninja Tabi for auto-attack reliant comps, or Mercury Treads if the have a lot of CC.
For example: 
and round it off with
or
.
As for summoners, you don't take
because it doesn't add any combat power, and minimal map control power. If you want it to add power, that takes up an item slot, so it's just not worth it. You'll probably want to go either
or
, as well as obviously
. And yeah, I think that covers everything. Best of luck.