I'll pipe up.
Attack damage, attack speed, and armor penetration all scale better the more you have of the others. This means that if you have a lot of attack damage and attack speed, armor penetration will increase your damage output the most. In some cases, attack speed does not help certain champions scale. These champions, such as Pantheon and Garen, deal damage primarily through their spells. As such, they benefit the most from arpen and AD.
With the basic definition out of the way, arpen is best when you're going for a mid game power spike. The reason for this is that most champions who build AD/arpen have low base stats to begin with, and you still need the other two stats to scale effectively. So, if your first item is going to be a Tiamat, it's a good idea to take arpen in your runes, since that allows for the biggest increase in damage output. The same will go for the opposite. If you have raw AD in your runes and masteries, a Brutalizer will increase your damage output more. An example of where I would take armor penetration in my runes is when I play Jarvan. He usually builds a Tiamat in lane, and he gets a huge power spike at six when he gets his ultimate, which has high base damages and is physical damage. Personally, I take armor penetration marks in the jungle and in lane when I'm playing AD based champions, since I nearly always build a Tiamat first.
I hope that helps, and I'll check back if you need anything else.
EDIT:
I forgot to mention that the game prioritizes armor penetration in different ways. A Black Cleaver will not influence your DPS in the same way a Brutalizer will, for example. Percentage armor reduction and penetration take precedence in that order. Then, flat armor reduction and flat penetration. In other words, a Black Cleaver will reduce the total amount of armor that a Last Whisper will ignore. It makes Last Whisper more inefficient. Keep this in mind when considering your damage output options.