Wow, to make it VERY simple...
Laptop power supplies have a 110AC to DC converter inline. Meaning you have a power cord that plugs into the laptop, goes to the inline converter and a 110 plug on the other end.
The car adapter one is basically the same thing, just that you supply it with 12 volts instead of 110.
So get a 12v power supply for the laptop, simply plug it into the Cig lighter (Or as mentioned, just cut off the Cig Lighter plug) and plug the other end into the EXISTING laptop power plug.
Laptop (o)---------[12v Laptop power supply for your model laptop]----------[ 12v power from car.
It just could not be any easier