What exactly do the items Last Whisper and Void Staff do?

Both Last Whisper (+ Lord Dominicks Regards and Mortal Reminder) and Void Staff give a percentage of Magic Penetration and Armor Penetration.

What does this mean? The wording on the Last Whisper (and the items that build from it) say X% bonus armor penetration. Does this mean it gives me X% of the armor penetration I have? Or does it ignore X% of my opponent's armor total?

Every champion in League of Legends has armor and magic resistance stats. Even marksmen, who typically don't build any defensive items other than Guardian Angel or Mercurial Scimitar, have some base resistances that reduces damage.

When you buy defensive items, you are often getting additional armor or magic resistance stats. This additional value is called bonus resistances.

Last Whisper (and its subsequent items) penetrates only bonus armor. That means that base armor is ignored.

For example, using completely fictional numbers, let's say we have a champion that has 10 base armor that has purchased 90 armor, for a total armor stat of 100. Last Whisper will reduce his armor rating by 27 (30% of 90), leaving his total armor stat at 73. As another example, a champion with 90 base armor and 10 bonus armor will only have his armor rating reduced by 3 (30% of 10), leaving his total armor stat at 97.

Void Staff is a little different - it ignores 35% of the total magic resist. So using the two sample champions from before, no matter what their ratio of base vs. bonus magic resist is, they would both have their 100 total magic resistance changed to 65.