While there are a lot of chemical combinations that can and have been made into useful batteries, in practice there are only four rechargeable types readily available in sizes suitable for ebikes. These are Lead Acid (PbA), Nickel Metal Hydride (NiMH), Nickel Cadmium (NiCad), and Lithium-ion or Lithium Polymer.
For a long time, lead acid has been the defacto standard for EV's. The cost is low and the chemistry well understood: Always charge up the lead acid battery whenever you can, never leave it in a flat state, expect only 60-70% of the rated amp-hours, and be glad if you get 200 cycles in a deep discharge environment. Probably 80% of all ebikes sold around the world still use lead acid battery packs, but their days are limited. The weight of lead needed to propel a bicycle for a decent 40-50km range is simply too much for a bicycle to easily handle.
Nickel Cadmium was the old standard for rechargeable consumer cells in the familiar AA, C, 9V series. They are known for robust characteristics, a good cycle life, and high discharge capabilities. They are still widely used in cordless power tools, R/C toys and similar applications that demand large currents, but for nearly everything else NiCad's have been replaced by NiMH and Lithiums.
Nickel Metal Hydride is quite similar to Nickel Cadmium, but with a higher energy density and a safer environmental record when disposed of in landfills. This is the dominant rechargeable battery type in digital cameras and other consumer products that offer user replaceable cells.
Almost all consumer electronics that have a plug-in charger these days are powered with lithium batteries because they can store about 3 times more energy than NiMH. Small devices like cell phones, mp3 players, and other gadgets typically have lithium-polymer packs, as these can be formed in conveniently shaped thin rectangular pouches. Larger devices like laptops and the new lithium cordless power tools generally use cylindrical Lithium-ion cells of a size smaller than a 'C' but bigger than a 'AA'. These are spot welded in series/parallel combination to give an appropriate voltage and capacity for the job.