How is density used to determine if your car battery is good?

The electrolyte in a car battery is sulfuric acid, diluted with water. The density of the acid when fully charged is 1.265g/cm3. The chemical process, as the battery is discharged, makes the acid weaker and it becomes less dense. Down to 1.120g/cm3 when below 11volts.
Checking the acid density will give you an idea of the state of charge.
There are 6 cells in a 12v car battery. The charge of each cell should be the same. If the density is significantly different between cells, then that would indicate a fault in that cell and the battery is useless. Faulty cells may have distorted and shorted plates which would stop it being charged and show up with a different density than the other cells.