Stoves generally break into two categories: white gas and butane/propane. White gas stoves have a metal fuel bottle you refill with camping gasoline, and these generally burn hotter than canister stoves. You usually need to prime the stove, which adds time to the cooking process.
Canister stoves are simpler to light, but generally no recyclable. Fuel is contained in a sealed cylinder that you toss out when empty. Some downsides include more expensive fuel, and more waste. But canister stoves are extremely lightweight and easy to use. Both options are very lightweight and work quite well for most backpacking trips.
Alcohol stoves are the lightest weight and easy to use. Again weight is everything.