The main idea:
The basic purpose of them is the protection from rockets and ballistic weapons. They consume a lot of energy and are ineffective against lasers. This is their weakness. 
How they work:
Each shield has parameters: shield radius and power use. All modules, which are inside the shield do not get damaged until the shield is depleted. 
The meaning of "Shields strength" shows how much damage this shield can reflect before it's power is gone. 
If the shield has the power after damage, he recovers. The speed of recovery is pointed in the "Regeneration Speed". The max power recovery is pointed in "Max Regeneration."
You have to know the shields may have not enough the power to stop the enemy if your enemy has strong weapons.  In this case, when the shield loses its power, it switches off. 

Are the shields needed for every ship?
Yes, the more shields, the more safe your ship is. But pay attention to the size and weight of each shield. And don't forget that lasers will totally ignore your shields!