Three types of cooking burners are commonly in use.
Hexamine burners are a staple of army ration packs. The burner itself is a light aluminium frame. The fuel consists of blocks of wax-like hexamine. The advantages are that they are light, are easily lit, and burn at a reasonably consistent but non-changeable temperature. The disadvantages are that while they can boil water OK, they are mainly suitable for heating food as quickly as possible and not really designed for preparing 3 course meals. Oh, and they make the bottom of the cooking containers sooty. I like to keep one in my webbing just in case. As a quick and dirty meal, simply dent or put a whole in a can of cat food, oops – I mean Irish Stew – and put it on the burner. The dent/hole is necessary so hot gas doesn't lead to you getting hot food sprayed into your face when you open a heated can.
Undoubtedly one of the commonest fuel sources, butane (sometimes propane – I'll use butane as a shortcut for both but for interest's sake, propane is more expensive and burns hotter) is commonly sold as disposable canisters such as the Campingaz, MSR, Primus, and a range of other brands, with burners in a range of configurations and sizes. While undoubtedly simple and easy to use, they pose some problems for the hiker. First, although they are light, the burners and fuel are bulky. Second, because they are pressurised containers, temperature differentials between the atmosphere and the container alter the burning ability. As the cylinders are used, they cool down due to gas expansion - a general phenomenon of gases. The flame will die down (see One Burner Cooking). Most of these canisters are pierced by the burner on first use, so you have to use them until they peter out. This problem can be avoided by using canisters that can be swapped, and kind of juggling them around during cooking. It becomes easy with practice. The canisters can be expensive, and in some places finding your particular brand can be tricky - they are only occasionally interchangeable. I've been caught out before.
Refillable canisters generally have similar issues – bulk, and variable burning temperature although to a far less degree than the disposables because they are larger and contain more gas. They are, however, fillable at many places such as gas stations.
In short, if you don't mind carrying the bulk, butane is cheap and easy. They are great for car-camping, but a bit of a pain in the proverbial for hiking.
Liquid fuel stoves
A serious hiking stove would be a liquid fuel burner such as the MSR range. The one pictured is kind of cool as it can use both canisters and liquid fuel. They can be expensive. You can pick up other branded equivalents for less. I carry a more basic model MSR stove for hiking, with a hexy stove as back up for a convenient quick heat.The burner itself is compact, and extra fuel is kept in a nicely sealed container. As the fuel is liquid, not gas, the fuel is far less bulky than the butane equivalent. Fuel is pressurised by pumping by hand, so the problem of losing pressure due to cooling and/or freezing is less of an issue. These stoves are less affected by altitude or outside temperature.
The choice of fuel is varied. Methylated spirits is the least volatile of the range. As an aside, you can get burners specifically for Meths. I don't find them hot enough. But meths burns cleanly, albeit coolly, with less chance of spilled fuel causing unwanted fires. White spirit (Naptha) is a fuel of choice. The stoves can also burn kerosene (which gets a bit sooty) and gasoline (which makes some folks a bit edgy). They tend to have only two temperatures – Off and Hot-As-Hades. So be careful not to burn your morning crepes. Liquid fuel stoves take a bit longer to prime and start, and stay hot for a while so be careful when packing up.
Most national parks have serious fire restrictions, so carrying cookers in and being self-sufficient is, in my mind, far preferable and responsible. There may, however, be occasions when you need to use open flame. I consider this separately under survival.
It pays to keep this simple. If you are eating tinned food, simply dent or crack the tin and plonk it on the stove. Otherwise, a simple Cups, Canteen (to use the army vernacular of Backwards, Writing) fits neatly around an army-style water bottle and takes next to no space. This plus a spoon and a knife can fulfil any 1-pot cooking and eating needs. For those of you who really need to fulfil the Julia Child within you, there are a range of stackable kitchen options. On the positive side, they enable you to express your culinary talents. On the down side, they take a lot of space and unless you have lots of fuel to burn, you are unlikely to need everything in them.
For US Amazon go to Search and buy stuff: US and the rest