Armor, Shields, and Barriers are all "Defenses" that protect enemy (or Shepard's) healthbars from harm.
For Shepard personally, its largely irrelevant. Only two of the six possible Shepards have barriers (Vanguard / Adept), and none of the six have armor. For Shepard, Barriers and Shields differ only in color (purple vs. blue), as any armor piece that improves shields will likewise improve barriers.
Shepard cannot get armor at all; the Fortification power, despite claiming to be armor, is really a damage reduction, like Tech Armor (which, despite the name, is actually advanced shield tech).
For enemies, it matters a little more.
Armor
Indicated by a yellow health bar. Vulnerable to fire attacks, like Incinerate, and damaging biotic attacks like Warp and Reave.
Common to Cerberus Robots, Environmental obstacles, and reaper forces.
Slow firing weapons like Sniper Rifles, Heavy Pistols, and Shotguns tend to do better damage to armor.
Finally, Armor can be "weakened" by freeze effects and "penetrated" by weapon mods, or Armor Piercing Ammo. Incendiary Ammo, Warp Ammo or Armor Piercing Ammo, will all do additional damage to armor.
Barriers
Indicated by a purple health bar. Vulnerable to damaging biotic attacks like Warp and Reave, and by Concussive Shot.
Common to biotic users.
Fast firing weapons like SMGs and Assault Rifles do better damage to barriers.
Barriers also take additional damage from Warp or Disruptor ammo.
Shields
Indicated by a cyan health bar. Vulnerable to tech attacks like Overload and Sabotage.
Common to Geth, Cerberus, and certain Reaper forces.
Fast firing weapons like SMGs and Assault Rifles do better damage to Shields.
Shields take additional damage from Disruptor ammo.
Power recharge doesn't affect Nova since whether or not you can use Nova depends entirely on if you have shields or not. The biggest point of the power recharge is to make your Biotic charge have a lower cooldown so you can go into a crazy Charge->Nova->Charge->Repeat cycle until whatever you are targeting dies.
If you are comparing absolute DPS, the Charge->Nova->Charge->Repeat combo can't be beat. The reason behind this is that, once your power recharge rate goes up high enough, your charge will be instantly available right after a Nova, and rather than doing a second nova at half damage, you can do a full blown charge at max damage. Since this is the case, you want your only nova to do as much damage as possible rather than just 60%.
However, if you worry about survivability, then you can go for the half powered nova so you retain some shields.
In my first playthrough, I went the entire game using the Charge->Nova->Charge pattern (difficulty hardcore), and I had no issues with survivability. Just don't do something silly like Charge into a banshee and get impaled.
Best Answer
Like in Mass Effect 2, each individual weapon has different base damage values against Health (Red), Armor (Yellow), Shields (Blue), and Barriers (Purple).
The information linked to in the post is true for the general case - most Pistols deal the most damage to Armor. But you have variance between the individual weapons as well - the Arc Pistol is a pistol that specializes in dealing maximal damage to shields, not armor.
The exact benefit against any given defense depends on the individual weapon. (Melee and the various heavy melee abilities can be considered as weapons in this case)
That clarified, Armor, Shields and Barriers provide absolute protection until they fail (they are effectively a separate health bar), barring special abilities suck as the armor-piercing weapon mods (which allow some damage to 'penetrate' armor), and armor piercing ammo (same).