Of the four, Lithium Polymer is the latest and most advanced type of cell phone battery. The distinguishing features of this type of battery are light weight technology as well as longer battery life. They are also smaller in size. For their weight and size it is difficult to beat Lithium Polymer batteries performance. They can last almost twice as long as lithium Ion batteries. Nickel Metal Hydride batteries were introduced to the market roughly 20 years ago. They are reliable and can be recharged to full within a few minutes. The down side is that they do tend to have a memory effect and tend to be heavy. However, they tend to keep the charge for long periods of time and are relatively cheap compared to the rest.
Lithium Ion or Li-Ion batteries are the most common and popular cell phone batteries on the market today. This is because they give the most 'bang for the buck' compared to all other types of batteries. They are very reliable, hold a charge well, compact and do not suffer from memory effect like some of the other batteries. In short, if you cannot afford Lithium Polymer batteries then these are the ones to go for. Most online as well as brick and mortar stores will carry these.
Nickel Cadmium batteries are the oldest of the lot. The main problem with these batteries is that they have serious memory loss problems. This means the charge needs to be completely used up before recharging. If not, then there is always the possibility of damaging the battery. The result will be longer charging times. The technology used to make these batteries is quite old and outdated. Hence they are more hazardous to the environment compared to other cell phone batteries. This brings us to the last point and that would be cell phone battery disposal. It is always better to give it to a battery disposal center, rather than just throw it away after its life.