It is fairly uncommon to see a deep cycle car battery, because they are less suited for use in automobiles than traditional car batteries. You are much more likely to see a deep cycle battery in a boat or golf cart than a vehicle with an internal combustion engine. The following will give you all the information you need to understand how deep cycle car batteries work.
Like any lead acid car battery, a deep cycle battery works by creating a chemical reaction in a sulfuric acid solution, called electrolyte. There are two metal plates inside the battery: one is connected to the negative terminal and is made of lead, while the other is connected to the positive terminal and is made of lead dioxide. When the battery discharges, the lead is excited by an electric charge and it reacts with the sulfuric acid to create a lead sulfate ion with an extra electron. The lead dioxide plate can then react with hydrogen ions and sulfate to make more lead sulfate and water. As the battery discharges, the plates build up lead sulfate and the electrolyte gets more diluted. In cars, the battery can be recharged using an outside power source, such as an alternator, to reverse these chemical reactions.
The difference between deep cycle batteries and more traditional lead acid car batteries is that deep cycle batteries use much thicker lead and lead dioxide plates. This diminishes the surface area on which the chemical reaction can occur. It gives the battery the ability to discharge over and over, but diminishes its ability to produce a lot of power over a short time.