Shields are stronger enemies' basic defense. Only after you destroy them you can deal physical damage to an enemy. Some of them are clever enough that when their shields are depleted they run away and hide because after a short while their defense regenerates. Because of that destroying shields from a long distance is not a good idea.
A blue energy shield of a mini-boss
Shields can be destroyed by using any of the weapons, however, you should choose them accordingly to shield's color - there are three types of shields: orange - Solar, blue - Arc, purple - Void and white. As you have probably already noticed, some weapons in your inventory are marked in that way. Select a weapon with the right markings (that is, a weapon with blue icon should be used on blue shields) and thanks to that you can, after only a few shots, get rid of enemy's shield thanks to additional bonus damage. Kinetic weapons deal the least amount of damage. Try to use energy weapons (a color different than the one required) against enemy shields and you will still get a small bonus to your attack. In the beginning of the game you should focus on only using weapons with orange and blue icons, purple ones are hard to find at that point.
White shields are different. Usually, they are barriers that grant complete immunity to damage to that enemy. When you come across opponents like that you won't see any damage numbers when you shoot them. In most of the cases those enemies are supported by others that give them those shields - destroy the support first and then deal with the main enemy. Usually, those shields are given by Servitors and Psions.
There is an additional type of shields, those that are held in hand (e.g. Phalanx). They grant complete immunity to damage. If you want to get rid of those persistent enemies you should try to move to their backs by jumping over their shields or you can do something even better - attack them when you are in the air - or by shooting at their legs (however, this deals a small amount of damage). If you have enough skill and experience you can try to throw a grenade in a such way that it lands behind an enemy.