Rechargeable Battery Types - definition
Cellphones of the past came equipped with rechargeable nickel-based batteries with (Ni-Cd and NiMH).
In the past few years, however, most rechargeable batteries used in phones are Lithium-based.
They are either Lithium-ion (Li-Ion) and Lithium Polymer (Li-Po).
The fundamental difference between a polymer lithium-ion battery and a liquid lithium-ion battery is that the electrolytes used in the two are different.
Lithium-ion batteries have high energy density, so they hold more charge per volume but lose charge with aging even when not used.
Li-Polymer has a lower energy density battery, but they are thinner, lighter, safer, and retain charge better as their age.
Lithium polymer batteries are more advanced and significantly more expensive than Lithium-ion to manufacture so neither is definitively better than the other. Manufactures usually decide which type of battery is better suited for the the particular device design.