Looking around on the GuideScroll Warframe guides I have found the following information:

GRINEER: Takes bonus damage from Armor Piercing.
CORPUS: Takes bonus damage and gets staggered by Electricity.
INFESTED: Takes bonus damage and some gets staggered by Fire, some are immune to Electricity.

Freeze damage dose 200% damage to shields where all other damge type do 100% damage.

Most enemies will recive a bonus resist based on there level that is subtracted from your damage.

Damage is listed as Normal hit/Weakpoint hit, for most that is Body/Head.

GRINEER

Everything is armored except for melee Grineer.
Normals include trooper, Lancer, etc.
Heavy include Bombadier and Gunner.
Shield Grineer takes damage as Normals, but recives 2 levels less resistance.
Commander-normal is the same as Heavy but immune to electrical/fire damage, AP/freeze not tested.

        Normal      Armor Piercing  Fire        Freeze      Electrical
Sawmen  100%/300%   100%/200%       100%/200%   100%/200%   100%/200%
Normals 33%/100%    150%/300%       37.5%/75%   16.5%/33%   20%/40%
Heavy   66%/200%    100%/200%       50%/100%+   50%/100%    50%/100%

CORPUS

Crewmen are armored.
They are Staggered by Electricity.
Moa's weak point is there midsection/gun.

        Normal      Armor Piercing  Fire        Freeze      Electrical
Crewmen 81%/0%      100%/2000%      33%/100%    75%/200%    160%/400%
Moa     100%/200%   50%/100%        50%/100%    50%/100%    200%/400%
Osprey  100%        100%            100%        100%        200%
Cameras 100%        100%            100%        100%        200%

INFESTED

Normals include Chargers, Leapers, Runners, etc.
Ancient heads/bodies and Crawler bodies are armored.
Normals and Ancients are Staggered by Fire.
Crawlers have 3 areas of damage Body/Legs and Arms/Head.
Ancients have 3 areas of damage Body/Head/Lower legs and Arms.

                    Normal          Armor Piercing  Fire            Freeze          Electrical
Normals             100%/200%       50%/100%        200%/400%       100%/200%       50%/100%
Nauseous Crawler    66%/100%/200%   50%/50%/100%    125%/200%/400%  50%/100%/200%   50%/50%/100%
Crawler             50%/100%/200%   50%/50%/100%    100%/200%/400%  50%/100%/200%   50%/50%/100%
Ancients            25%/75%/100%    100%/200%/100%  37.5%/75%/200%  25%/50%/100%    0%

If you're here from the future, the damage calculator pointed out by Oblivious Sage is outdated as of Damage 2.0. Warframe's wikia site has an up-to-date, extensive paragraph on damage calculation, along with formulas.

The system is quite complex, but the gist is:

• Enemies (and NPCs, and warframes) are made of "health materials" - ie. different types of health, shields and armor.
• Every "health material" applies different multiplier against different damage types.
• When you attack, the final damage within your attack is calculated separately for each damage type.
  • Health is a plain, old health bar. If it's depleted, the enemy dies.
  • Shields is a separate health bar that serves as a protection for health bar. If it's depleted, the damage goes to health.
  • Armor is not a health bar. It adds its own multipliers on top of health's multiplier (cumulatively) and reduces damage the health takes. However,
    • the more armor the enemy has, the more weight armor's multiplier has over health's multiplier,
    • damage types strong against armor reduce that protection (dealing more damage), while damage types weak against armor increase that protection (dealing less damage) - on top of the damage multiplier.