Once the shields have been drained from a certain area, any damage to that area results in damage to the hull. Most shields have a "Bleedthrough" percentage, which is the amount of damage from any enemy's weapons that cause damage to the hull instead of the shields. Your current Damage Control skill determines the passive hull regeneration rate of the ship.
The amount of hull depends on the tier and type of starship, with Cruisers and other bulkier ships generally having the most hull, while Escorts and other more fragile ships have the least, though there is a wide variety—see the Playable starship page for a rundown of each ship type. The hull strength of Small Craft scales with the player's level, up to level 50—see Small Craft § Hull scaling for details. Similarly, the hull strength of Tier 5-U and Tier 6 starships begin scaling at level 50—see Playable starship § Tier 5-U and Tier 6 hull scaling for details.