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.
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.
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.
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.