Primaries

Single-use cells are often referred to as "primaries" or "disposable." Primary cells are not rechargeable, secondary cells are.

Carbon-Zinc

1.5V/cell. Also marketed as "Heavy-Duty." These do not last as long as alkaline batteries and do not tolerate heavy loads at all well. Tend to leak when asked to produce more than a few tens of milliamps and when exhausted. Are not able to power devices requiring large current such as digital cameras or flashguns.

Alkaline

1.5V/cell. These will last several years on the shelf. They have higher internal resistance than NiMH rechargeables and lithiums, so they do not perform as well with high-drain applications. If they charge down too far, they are prone to leaking and corroding. They can be very economical in low-drain uses like clocks and remote controls where they will last for years without needing replacement. There are different levels of performance available and Duracell, for instance sells an Ultra Advanced version that has more capacity than their regular copper tops.

Nickel Oxyhydroxide

1.7V/cell (?). Duracell PowerPix batteries use a nickel oxyhydroxide (NiOx) chemistry that is supposed to last twice as long as alkalines in digital cameras (high drain) and has a higher voltage, similar to lithiums. However, the actual performance does not seem to be that much higher than alkalines and in low and moderate drain use, there is no advantage.

Lithium

Important: ***These are not the same as lithium-ion batteries that can be recharged. These are "primaries" and therefore can be used once and then recycled. If you attempt to recharge them, they will explode (example).***

Lithium batteries have a very long shelf life. After 10 years, they might retain 80% of their original charge. They are a good choice for an emergency flashlight or one that is not used that often. Because of the extra capacity, some people use them to get the longest possible runtime. They also weigh noticeably less than other batteries.

Energizer - 1.7V/cell. The higher voltage makes some flashlights brighter than on alkalines or NiMHs, briefly. Energizer retains a patent on lithium AA and AAA batteries in the US and therefore are the only company to offer this chemistry there. These batteries are widely available and come in two varieties: "advanced" and "ultimate". The "ultimate" has blue on the label and is the same thing as the earlier e2 lithiums (both versions have the same part number: L91 for AA and L92 for AAA batteries). These are advertised as lasting 8 times as long as alkaline batteries (realistically it might be 4 times as long, but they cost 3 times as much). The "advanced" lithiums have yellow on the packaging (the part numbers are EA91 for AA and EA92 for AAA cells) and are advertised to last 4 times as long as alkalines, but in practice they have about 75% of the capacity of Ultimates (so maybe 3 times as long as alkalines). They are usually less expensive than the Ultimate Lithiums. While they have an initial voltage in excess of 1.7V, this bleeds off very quickly and the temporary increase in brightness compared to alkaline cells is short. Their real advantage is very long shelf-life and low-temperature tolerance, they will still produce useful current at lower temperatures than just about all other cell chemistries. Otherwise they are extremely expensive for the capacity they offer.

CR123A and others specialty cells are also lithium and usually 3 volts per cell. There was a time when some made in China CR123A cells were causing problems for people on CPF and therefore some will only recommend buying Made in USA cells which never displayed those problems.

Saft - This company has a line of lithium primaries (LiSOCl2 - Lithium-thionyl chloride chemistry) that are 3.6 volts per cell.

Button Cells

Some very small lights and keychain lights use button cells. These come in a wide variety of sizes and chemistries which are explained in greater length in the Wikipedia article.