For starting fires in our campsite, I use dryer lint and wax for tender (see How to make your own campfire starters) which is a quick and easy way to get a campfire going. Anytime I venture out from the campsite, however, I carry a number of emergency items in a small daypack, including a fire starter kit.
An emergency fire starting kit needs to be small and lightweight, otherwise you are likely to leave it behind. It also needs to be ultra-reliable, since in the extremely unlikely situation that you would ever need to use it, you will probably be cold, possibly wet, perhaps injured, and probably not in your best state-of-mind.
My emergency fire starting kit consists of cotton balls smeared with a thin coating of petroleum jelly, that I store in a small Nalgene® bottle, and a magnesium sparking tool from Swedish FireSteel®. I have found that this type of setup to be virtually “Murphy proof” and extremely reliable under all kinds of different conditions. Particularly when your hands are cold and numb, the Swedish FireSteel is a lot easier to use than trying to strike a match and the petroleum-smeared cotton balls burn hot enough, and long enough, to ignite even damp kindling.