Maximizing Lethality and % Armor Penetration
First, I am starting off explaining what these things are, then I will explain how they relate, and which is better above or below their convergence point (where one helps EXACTLY as much as the other). Feel free to skip to the different sections if you feel you already know a section, and would rather just get to the part you didn't know. The table of contents is below
o Explanation of Armor, and what it does o Explanation of % Armor Pen, and how it works o Explanation of Lethality, how it works, and how it came to be o Comparing Effectivity of Lethality and % Armor Pen (which is better at what point, etc.) o Additional Bits and Pieces
Explanation of Armor, and what it does
First, the significance of armor can be seen by hovering over the armor statistic typically found around the bottom-left of your screen. When you hover your cursor over it, it will say "##% Dmg Reduction" (some number percent PHYSICAL damage reduction). The more armor you have, the higher the damage reduction. This is not a 1% dmg reduction per 1 armor. The actual equation is: ##% physical damage reduction = armor / (armor + 100).
So, if you get 6 Thornmails 
and have 80 base armor (the armor you get merely for being that champion), you receive 87% less damage from ALL incoming physical damage, that is, unless the enemy has lethality or armor penetration.
Malphite has a passive on his W ability that gives him a percentage more armor, and at level 18 he has 92.05 armor just for being Malphite (not adding the two together yet). If Malphite got 6 Thornmails, he would receive 90% less damage from all incoming physical damage. For 934.2675 total armor... (242.2175 more armor than most other champions with that build) that is not as much as one would expect. After all, an additional 242 armor added a wonderful... 3% damage reduction.
With this in mind, excluding lethality and % armor pen, having more than 230 armor is of little use, having approximately 70% physical damage reduction at that point.
Explanation of % Armor Pen, and how it works
Now, percent armor penetration (% armor pen) is useful to cut through high armor enemies (such as that 230 armored one). It is useful because it eliminates a percentage of the enemy's armor. This changes the equation from: % physical dmg reduction = armor / (armor + 100) to % physical dmg reduction = (armor - (armor x % armor pen)) / ((armor - (armor x % armor pen)) + 100).
Example: if you have 250 armor, this will be your damage reduction: 250 / (250 + 100) = 71% damage reduction. But, if they have Lord Dominik's Regards 
, at 45% armor penetration: (250 - (250 x 0.45)) / ((250 - (250 x 0.45)) + 100) = 137.5 / (137.5 + 100) = 58% damage reduction.
By getting that ONE item, they now deal 13% more physical damage to you than before, in addition to the additional attack damage, and bonus damage if you have more maximum health than they do.
Note that just as armor becomes less effective the more armor you get, armor penetration is ALSO less effective the more armor you get. Let's go back to that 6 Thornmail idea again.
With 680 armor, you can expect 87% reduced physical damage. If they get , you still expect 79% reduced physical damage. They now deal 8% more physical damage to you than before. That is certainly less than 13%, but still. Now, for Malphite...
With 934 armor, he can expect 90% reduced physical damage. If they get , he can still expect 84% reduced physical damage. They now deal 6% more damage to him.
Explanation of Lethality, how it works, and how it came to be
Lethality is designed (generally) for physical damaging assassin work -- rushing in, getting a kill, and rushing back out before the enemy even realizes you are there. Lethality is more accurately described as flat (not percentage) armor penetration, increasing as your enemy levels up. If your enemy is level 1, you have less flat armor penetration than if your enemy is level 18. When I say flat armor penetration, I mean 1st, this is NOT percentage based, and 2nd, the value of your (or the enemy's) armor is reduced. If you have 10 (flat, so this is just 10, NOT 10%) armor penetration, then the enemy you are damaging is affected as if they have 10 less armor. If your enemy has 80 armor, and you have 10 armor penetration, it will be as if they only have 70 armor. (Keep in mind, lethality does not increase your ally's physical damage to the enemy, only your own.)
As mentioned previously, the flat armor penetration of Lethality increases as your enemy levels up, there is an equation for this: http://boards.pbe.leagueoflegends.com/en/c/champions-gameplay-feedback/RYuvnjbT-lethality-info . That is a link to the page of a RIOT Employee explaining some about Lethality, including its equation. According to the RIOT Employee, this is the Lethality equation: [Flat Armor Penetration] = 0.4[lethality] + ((0.6[lethality] x [level of enemy champion]) / 18). If you have 20 Lethality, then, at (opponent) levels of 1, 9, and 18, you have Flat Armor Penetration against them of: 8.6, 14, and 20. Example: if you have 20 Lethality, and your enemy is level 9 and has 100 armor, it will be as if he only has 80 armor. If we add in the armor equation, that means: with no Lethality, your enemy will receive 50% less physical damage from you, and with 20 lethality, your enemy will receive 44% less physical damage from you.
Note that this flat armor penetration extremely quickly loses value as the target gets more armor. The 6 Thornmail Malphite would notice about a 0.01% damage increase if you have 20 armor penetration, maybe even less.
Armor Penetration did not used to be so confusing, lengthy, and difficult to process -- it used to simply be called flat armor penetration, and acted just like flat armor penetration, namely, you (if you have the armor pen) deal physical damage to them as if they had that much (amount of flat armor pen) less armor. This changed because physical damage assassins were too strong starting out, or too weak in the end. Starting out with armor penetration when champions' initial armor is low, and nobody has any extra armor would cause huge amounts of bonus damage, resulting in the assassin getting super strong, super early in the game. OR, if the assassin is behind early in the game, the assassin stayed behind... all, game, long. This is what is called snowballing (either rolling into one HUGE success, or rolling into one HUGE failure). They then decided to create Lethality to replace the original flat armor penetration. Lethality builds up armor penetration as the game progresses, so the physical damage assassins no longer have a super strong advantage at the start of the game, and at the end of the game (when all of your opponents, and probably all of your allies too, are level 18) Lethality is just another word for flat armor penetration with 1 [lethality] giving you 1 [flat armor penetration]. While doing this, they also made assassins a little stronger at the end of the game too. Lethality is more available now through items and runes than flat armor penetration had been; additionally, Lethality is generally higher than the flat armor penetration had been (10 flat armor penetration would now be 15 lethality -- late game gives 5 additional armor pen). This (mostly) ended the snowballing of physical damage assassins.
Comparing Effectivity of Lethality and % Armor Pen
First, a generic explanation of which is better: % Armor Pen is typically the stronger of the two; Lethality is most effective against the enemies that have the LEAST armor -- % Armor Pen for higher armored enemies and Lethality for lower armored enemies.
Lethality goes into effect AFTER % Armor Penetration. This means that if you have 30 Lethality and 45% Armor Penetration, and the enemy is level 18 with 200 armor, you will deal 55.6% of your outgoing physical damage to them (they have 44.4% physical damage reduction). Flat Armor Pen = 0.4[lethality] + ((0.6[lethality] x [opponent level]) / 18) = 0.4[30] + ((0.6[30] x 18) / 18) = 12 + 18 = 30 Flat Armor Pen ((armor - (armor x [% armor pen])) - [flat armor pen]) / (((armor - (armor x [% armor pen])) - [flat armor pen]) + 100) = ((200 - (200 x 0.45)) - 30) / (((200 - (200 x 0.45)) - 30) + 100) = ((200 - 90) - 30) / (((200 - 90) - 30) + 100) = 80 / 180 = 0.444 = 44.4% physical damage reduction.
Outgoing damage means the amount of damage you are dealing, without their damage reduction taken into effect. This includes abilities (and their ratios), item passives, critical strikes, etc...
Black Cleaver 
reduces enemy armor. I have been informed, through the comments below, that armor reduction is taken into effect BEFORE % armor penetration. There is additional information regarding this at the end of this section.
With all of this in mind, when trying to decide which is better for any given situation, when it is either one OR the other (to enhance your physical damage to them), it is useful to have a SIMPLE equation. This equation will identify that specific spot when % armor penetration would give exactly as much armor penetration as flat armor penetration. (Unfortunately, you have to determine how much flat armor penetration it is, as this does not directly include Lethality, unless the enemy is level 18). [Flat Armor Pen] / [% Armor Pen] Any armor higher than this, % Armor Pen will be more effective. Any armor lower than this, Flat Armor Pen will be more effective.
If you are going from the Lethality perspective... Level 1: (0.4[lethality] + (0.6[Lethality] / 18)) / [% Armor Pen] Levels 2 - 17: (0.4[lethality] + ((0.6[lethality] x [opponent level]) / 18)) / [% Armor Pen] (this one actually works for all levels, but the others are for convenience) Level 18: [Lethality] / [% Armor Pen] _(Useful and user friendly for late game, when enemies are all lvl 18) Levels 18+: (0.4[lethality] + ((0.6[lethality] x [opponent level]) / 18)) / [% Armor Pen]
Because % Armor Penetration applies FIRST, that enables Lethality to make a much more significant impact, so using them together is the best method. If you have 45% Armor Pen, it would be wiser to get 20 Flat Armor Pen instead of an additional 15% Armor Pen (like a Youmuu's Ghostblade 
instead of immediately finishing last whisper item (
): into either or 
) if they have less than 78 armor. (If you already have 15 Flat Armor Pen when you get 20 more Flat Armor Pen, then it would be less than 137 armor). When considering ADDITIONAL % Armor Pen over Flat Armor Pen (or additional Flat Armor Pen), you should use this equation. (Lethality to Flat Armor Pen not included)
([Current Flat Armor Pen] + [Additional Flat Armor Pen]) / ([Additional % Armor Pen] x (1 - [Current % Armor Pen]))
Note that if you are having issues damaging a champion with a high armor item, like Thornmail or Frozen Heart, or against a high armor champion, like Taric, Rammus, or Malphite, it would be better to finish the Last Whisper item BEFORE more lethality.
If the Black Cleaver's reduction comes first, as its own individual self (rather than simultaneously with the % armor pen from last whisper items), then things change a little should you choose to get it. Also, some champion abilities reduce armor on a flat and percentage base. I need to do further (in-(custom)-game) study on the matter.
Additional Bits and Pieces
The Maximum Armor Penetration (item only) build is:
Lord Dominik's Regards (or Mortal Reminder) or
Black Cleaver
Youmuu's Ghostblade
Duskblade of Draktharr 
Edge of Night 
and one other item. This build will cause a level 18 opponent with 400 armor to receive 77% of your outgoing physical damage, rather than 20%. That Malphite would receive 36% of your outgoing physical damage, rather than 10%.
I will add to this if I find something else to mention from the comments.
1st addition: I need to do some testing regarding armor reduction, particularly on black cleaver, and how it relates to everything else. If black cleaver DOES in fact, reduce the enemy's armor BEFORE % armor pen, then the above situation with malphite and the amount of physical damage you bring across to him is LESS than the 36% I have stated. (He has 75.6% physical damage reduction -- 24.4% of your physical damage gets across to him, instead of 10%.)
2nd addition: changes have been made to multiple armor penetration items (and masteries too). This is currently outdated. If you have read this, please realize that things have been changed since this was made, so until adjustments are made, please note that things need altered BEFORE it is completely accurate again.
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