Lead Acid - The original rechargeable. The same thing that's in your car. Low energy density (you need a big battery to get lots of power) Used mostly for belt starters.
Nickel Cadmium (NiCd) - The RC battery you grew up with. NiCd's are cheap and have higher discharge rates than similar NiMh batteries. They are heavier than NiMh and lack the capacity.
Nickel Metal Hydride - (NiMh) - The most popular battery in RC. A direct replacement for NiCd's Available in larger capacities (up to about 5200Mah)
Lithium Ion Polymer aka Lithium Polymer (LiPo) - not to be confused with lithium ion. The newest technology. 3.7 volts per cell. Very light weight. Very energy dense. Requires chargers specifically designed for LiPo's Can be permanently damaged if discharged below 3 volts per cell. More expensive than other types of batteries.