Ship Classes are different kinds of Ships. Each class and quality affects the ship in different ways - "light" ships are generally faster, but at the expense of cargo space and shielding. "Merchant" class ships have more cargo space, but are slower and less maneuverable.
The class a ship is can affect the shielding, the amount of upgrades it can recieve and the quantity of weapons.
The qualities a ship has can affect the cargo space, the acceleration and the maneuverability.