Liquid Fuel

In general, the most versatile of the stoves are the liquid fuel models.  They will burn a number of liquid fuels, including white gas (which I use pretty much exclusively in these stoves.)  White gas has a number of advantages over the other sorts of fuels.  White gas contains more energy than butane or alcohol, so over the course of a long trip, white gas tends to be more efficient than alcohol and most canister stoves (the exception are the Jetboil and Reactor systems.)  

White gas stoves are not adversely affected by cold as are canister stoves, and White gas stoves put out a lot more heat than most canister stoves (the Reactor is an exception.)   Lastly, white gas stove designs are typically good at all sorts of camp cooking.  You can use just about any sort of pot set, and the stoves perform well from full-blast to simmer, so you can melt snow, and fry up a brook trout using the same stove.  

I tend to choose a white gas stove for long trips where I am going to do a lot of cooking, frying fish, etc.  

I've used a lot of different white gas stoves over the years.  The two that I like the most are both Swedish made Optimus stoves, the Nova, and Explorer models. 

Cartridge Stoves

Cartridge stoves utilize a cartridge filled with a compressed mixture of butane and propane.  Cartridge stoves are all about convenience.  They are the easiest stoves to use; you just turn the knob, light them, and start cooking. 

Cartridge stoves have one serious drawbacks, however.  The Achilles heel of cartridge stoves is cold weather performance.  In cold temperatures, the butane gas is converted to a liquid state, and sits at the bottom of the cartridge instead of feeding vaporized butane/propane into the stove.  So, in cold temperatures, the stove's ability to function drops off dramatically.  There are some stoves (Like the Coleman X series) that address this cold weather problem by allowing the stove to draw liquid fuel.   

Hanging Stoves

Hanging Stoves come into their own on climbs, where there is very little space to cook.  Often, climbers find themselves on a ledge without a flat space to deploy a regular stove.  In nasty weather, hanging stoves are sometimes also used to cook inside a tent.  However, carbon monoxide poisoning is a very real and deadly danger when running a stove in the confined space of a tent. 

Alcohol and Wood Burning Stoves

I treat these two types of stove in the same section because they both tend to be quite minimalistic.  Alcohol and wood both put out relatively small amounts of heat, so the major design elements of a good alcohol or wood-burning stove are all designed to channel the available heat to the pot. 

Wood burning stoves take the most effort to light and maintain, and cooking times are relatively long because of their lower heat output, but their big advantage is that you need not carry fuel, so a wood stove system is very lightweight, and that weight advantage becomes greater the longer you are out on a trip.   

Alcohol stoves are simple and foolproof, as they use no moving parts.  Alcohol is also a relatively safe liquid fuel.  Unlike white gas, alcohol is not prone to vapor ignition over long distances and is therefore much less likely to explode or turn into an uncontrolled fireball due to a fuel leak.  (This has happened to me with a couple of white gas stoves.) 

The Jetboil PCS and MSR Reactor Stove Systems

Although these two stoves could be classified as both hanging stoves and cartridge stoves, they really deserve their own category.  These two stoves combine a stove with a pot that has an integrated heat exchanger that channels a very large proportion of the stove's heat into the pot.  Although windscreens and heat exchangers have been around for a long time, the Jetboil and Reactor systems take efficiency and function to a higher plane.