The main difference between models of camp stoves is the type of fuel they burn.

White Gas
Best Choice For: Cold conditions, high altitude, light weight
Pros: White gas stoves are inexpensive and available in lightweight models, making them a popular choice. White gas (also called naphtha or camp fuel) is cheap, volatile, evaporates quickly and leaves little residue.
Cons: The downside of these stoves is they often require a fair amount of set up (including a lot of manual pumping). They also require regular maintenance to clean off the burner.
Fuel Availability: Widely available in North America, Austrailia and New Zealand

LPG (Liquid Petroleum Gas - usually propane/isobutane/butane mixtures)
Best Choice For: Typical weekend backpacking or canoeing trips and car-camping.
Pros: LPG stoves are very cheap and because the fuel tank is pressurized, no pumping is required, making setup very easy - generally the only assembly that is required is to screw on the propane tank. LPG stoves range in size from classic multi-burner Coleman propane stoves ideal for car-camping to very small single burner stoves that attach directly to the top of the tank.
Cons: LPG tanks are a little heavier and voluminous than other options. Tanks are not refillable or easily disposable (though they are recyclable), so you'll have to carry empties around. Propane tanks last quite a while, but it can be hard to gauge how much is left, so you may end up unnecessarily carrying a spare tank.
Fuel Availability: Widely available in industrialized parts of the world.

Methanol
Best Choice For: Super light-weight trips.
Pros: Methanol fuel burns very efficiently and does not need to be pressurized to burn, making it an option for trips when weight is a big consideration.
Cons: Methanol is not great for quickly boiling a pot of water because it doesn't burn very hot. Many consider this a plus however, as it can be great for simmering a stew.
Fuel Availability: Available at most hardware stores in North America

Kerosene
Best Choice For: Bizarre destinations
Pros: Kerosene burns as hot as white gas, but is typically more widely available (often called Paraffin). Kerosene is also reasonably cheap.
Cons: Kerosene smells terrible and makes a huge mess if you spill it or a container breaks. Kerosene also requires a separate priming agent and gives off nasty fumes and soot that can contaminate your food so always put on a tight pot lid.
Fuel Availability: Available at most hardware stores and some gas stations.

Multi-fuel
Best Choice For: Just about anything!
Pros: Multi-fuel stoves can accept multiple types of fuel, making them ideal for traveling abroad, where fuel availability may differ.
Cons: Multi-fuel stoves are often expensive. The only other downfall is some models are difficult to convert between the different input options.
Fuel Availability: Available at most hardware stores and some gas stations.