Lithium-ion batteries have a number of important advantages over competing battery technologies.

They're much lighter than other types of rechargeable batteries of the same size.
They hold their charge. A lithium-ion battery loses only about 5 percent of its charge per month, compared to a 20 percent loss per month for NiMH batteries.
They have no memory effect, which means that you do not have to completely discharge them before recharging, as with some other battery chemistries.
Lithium-ion batteries can handle hundreds of charge/discharge cycles.

The electrodes of a lithium-ion battery are made of lightweight lithium and carbon. Lithium is a highly reactive element, meaning that a lot of energy can be stored in its atomic bonds. This translates into a very high energy density for lithium-ion batteries.

Here is a way to get a perspective on the energy density difference. 

A typical lithium-ion battery can store 150 watt-hours of electricity in 1 kilogram [2.2 pounds] of battery weight. 
A NiMH (nickel-metal hydride) battery can store perhaps 100 watt-hours per kilogram [2.2 pounds], although 60 to 70 watt-hours might be more typical. 
A lead-acid battery can store only 25 watt-hours per kilogram [2.2 pounds]. 
Using lead-acid technology, it takes 6 kilograms [or 13.2 pounds]  to store the same amount of energy that a 1 kilogram [2.2 pounds] lithium-ion battery can. That's a huge difference [Source: Everything2.com].
Lithium-ion also operate at higher voltages than other rechargeables, typically about 3.7 volts for lithium-ion vs. 1.2 volts for NiMH or NiCd. 
This means a single cell lithium-ion can often be used rather than multiple NiMH or NiCd cells.