It is a vessel containing a number of cells to produce a specific amount of electricity to power your laptop. Each cell contains different chemicals and the reactions between these chemicals produces electricity. Since each cell produces a specific amount of electricity, a number of these are required to provide the correct amount to power the laptop. These cells are packed together in a vessel, usually plastic, and hence laptop batteries are also known as battery packs.
As mentioned above this refers to the chemical composition of each individual cell. Since different chemicals have different properties this produces different characteristics on the cells and laptop batteries. The major types are as follows:
Nickel Cadmium (NiCd, NiCad)
This was one of the earliest type of laptop battery. It was very quick to charge and was able to handle high loads. As the name suggests it contains Cadmium, a heavy metal, poisonous and hazardous. Environmentally unsafe. This battery uses Nickel and Cadmium as its electrodes and aqueous Potassium Hydroxide as the electrolyte. This battery is synonymous with the "Memory Effect". This is where the battery remembers the capacity of the pervious charge cycle instead of it's original capacity.
Nickel Metal Hydride (NiMh, NiMH)
This uses Nickel and other rare earth metals as its electrodes and Potassium Hydroxide as the electrolyte. These are constructed from non-toxic metals making it much more environmentally friendly. These batteries do suffer from the memory effect, but not to the same extent as Nickel Cadmium batteries. They also have much higher capacity than the NiCad batteries. The life span of the NiMH batteries however is very short. Shorter than NiCad and Lithium Ion batteries.
Lithium Ion (LiIon, Li-Ion)
This is a newer type of battery. Compared to NiCd and NiMH batteries, Li-Ion batteries provide the same capacity inspite of being smaller and lighter. Li-Ion batteries have a higher power to weight ratio. Lithium Ion batteries use Lithium Oxide and a Carbon compound, usually Graphite as the electrodes. These are separated by a microporous film containing an organic solvent as the electrolyte. Li-Ion batteries do not suffer from the "Memory Effect" and have a very high Power to Weight Ratio. It also has low levels of self-discharging.
The latest type of battery chemistry is Lithium Polymer Ion. This behaves in the same way as Lithium Ion. However, it uses a thin plastic like film as the electrolyte. This dry polymer allows for a battery with considerably reduced weight. As a result Lithium Polymer Ion batteries have a greater power to weight ratio than most of the other types mentioned here. Due to this lightweight design their uses are ever expanding. Especially handheld devices. One of the uses of Lithium Polymer Ions is in Universal Laptop Batteries. Check the related guide on Universal Laptop Batteries.
If you want tips on getting the most out of your laptop battery then check this related guide: Extend Your Laptop Battery's Life.