Lithium Ion (Li-ion)
Lithium-ion batteries are becoming more popular for a number of reasons. Lithium-ion battery capcity is much greater than other rechargeable batteries, and lithium-ion batteries don't have some of the problems that older batteries, even rechargeable ones do. For one thing, they don't lose charge over time when not in use. They are small and compact, and yet produce just as much power, so they can be used in a variety of smaller electronic devices, such as cell phones, personal GPS, and iPods. Because they don't lose power until they are fully discharged, they are also ideal for keeping memory devices going until the very last minute, like digital cameras and laptop computers.
Just like any other type of rechargeable battery, lithium-ion batteries have their pros and cons. While lithium-ion battery capacity in electronic devices is much greater, the overall lifespan of the battery is fairly short, only lasting a few years. This means that while you can buy an extra battery if you need too, you should make sure that you use them a lot and well while you can. You must also expect that you will have to replace it and factor that into the costs when purchasing a new device that requires lithium-ion batteries. They also can't be exposed to temperatures above 120 degrees, so tropical climates, storing them in enclosed cars, and directly exposed to sunlight for long periods of time are not advised.
Lithium-ion batteries are much more delicate that other batteries so you have to treat them with care. For this reason you will not find rechargeable AA, AAA, C and D cells. Lithium-ion requires a sophisticated charging system that will not overcharge it as they may explode. Typically you will see lithium-ion batteries sold with laptops and cameras as they have their own built in charging circuits which are designed for that specific battery. Also, when removing or replacing a battery, be really careful not to drop it -- you don't want to start a chain reaction which will result in the lithium-ion cell reaching critical mass. Just kidding, but seriously don't drop them.
The lithium-ion battery life is long, and they will not form a memory if recharged before it dies completely, rather it is beneficial to do so. This is much different than a regular rechargeable battery. Letting lithium-ion batteries completely discharge too often or to let it sit uncharged for a long period of time will shorten it's overall lifespan. When storing them for a long period make sure they are charged to 50-60% and they shoudl be removed from the device they are meant to power.
If you notice a change in the lithium-ion battery life span, such as it not lasting as long as it usually does, then you should replace the battery pack. This is an indicator that at least one of the cells has died, and this cell could develop a short which could damage not only the other cells, but your electronic devices as well. Another thing to be watchful of as well. When lithium-ion batteries fail, they may do so in a dramatic way. They have been known to catch fire, explode, and even spew out noxious smoke. While internal recharging circuitry is designed to prevent over charging, you are the only one that can prevent this from happening from outside sources such as burning in a fire.