5:59 p.m. on June 24, 2011 (EDT)
Probably the most popular style of stove for backpacking these days is the upright canister gas stove type like the one Rick suggested, the MSR Pocket Rocket. They're light compact, and generally inexpensive. They do have a few drawbacks: they're "tippy" (high center of gravity, easy to knock your dinner off), vulnerable to wind, and poor performers in colder weather. The Pocket Rocket, Optimus Crux/Crux Lite, and Snowpeak Gigapower GS-100 are all examples of this type. You can also get the Jetboil which is a really fancy version of an upright canister stove. The new Jetboil Sol is a really nice stove system. Pricey, but really light and strong.
A step up, to my mind, is a remote canister gas stove where the burner and the gas canister are separate but connected by a fuel hose. These are a little more expensive, a little less compact, and a little heavier, but they're far more stable, work far better in wind, and most models can run in a lot colder weather, some below 0F. The Primus Express Spider, MSR Windpro, and Snowpeak Crab LI are all examples of this type.
Not as popular any more are liquid fueled stoves. These are able to operate year round even in the coldest of weather. Their price is more like a remote canister stove or maybe even a bit more, but in the long run they can save money because their fuel is so cheap. A four ounce canister of gas costs about $5.00 (+tax). The equivalent fuel for a liquid fueled stove? About $0.30. Liquid fueled stoves are a more environmentally sound type of stove: one refills the fuel tank rather than throwing it away. Some popular liquid fueled stoves include the MSR Whisperlite and the MSR Simmerlite.
There are also "alternative" (non petroleum based) fueled stoves:
Alcohol stoves which can be super cheap, particularly if you make them yourself, are typically very light, and are extremely quiet. They have the drawback of requiring more fuel (in terms of weight), and they are slow and vulnerable to wind. Trangia is a popular brand, but a great number of people make their own out of soft drink cans, cat food cans, etc.
Some people use wood stoves for backpacking. They're generally impractical in my area, so I won't comment on them.
Solid fueled stoves, typically hexamine which is sold commonly under the ESBIT brand name, are very popular with the ultra light crowd. These stoves burn little white cubes or tablets of fuel. These stoves have the most expensive fuel but are the very lightest weight option out there.
HJ