Lithium or Li-ion batteries are cheap, easy to use and recharge. There is actually no difference between Lithium Ion and Lithium Ion Polymer.
The advantage of the Li-ion is that the size is smaller, come in various sizes, actually they can be 1 mm thin and they are flexible.
There are 2 different types of Li-ion, the new one with graphite anode and one with coke anode (the traditional one). The advantage of the graphite anode is that the dropdown voltage is much smoother. The temperature of the battery will not be as high as the NiMH one in the charging process.
The charge uses the constant current – constant voltage process. The charging current is ~ 0.5/C, where C is the current capacity but it depends from manufacturer to other. For more details watch the video tutorial from EEVblog.