All characters possess shields called "kinetic barriers" that absorb a limited amount of damage. In order to harm an enemy, you must first blast through their shields. If shields are damaged, take cover and avoid further damage to give them a chance to regenerate. The Shield Boost power immediately recharges shields.
Some weapon mods do extra damage to shields, or allow shorts to partially bypass them to and strike the character within.
Armor protects characters from all damage that breaks through shields, reducing it by a fixed percentage. Each suit of armor also gives a level of resistance to biotic and tech power attacks. Heavier armor tends to provide greater protection against all attacks, but requires additional training and slows the character slightly.
Armor can have intrinsic special abilities. Some types will heal the wearer at a slow but constant rate. Others can protect the squad from conditions of extreme heat, cold, pressure, etc. on uncharted worlds. You can also gain these abilities by installing appropriate mods in any set of armor.
When you are in a habitable area, you can equip or un-equip your squad members' helmets on the Equipment Screen [ I ]. Removing helmets is purely cosmetic, it does not reduce protection from damage.