Ice will melt when heat is applied; paper will catch fire. In trying to figure out why — what the difference is between things that melt and things that catch — I found "Burn, Char, Melt" by Roberto Gregorius, who says it depends on
whether the substance easily forms compounds with some oxidant [usually oxygen]. This term "easily" can be taken to mean that the energy required to start the reaction (called the energy of activation) is low enough so that the reaction starts (as in striking a match), or the compounds produced in the burning are low enough in energy that there is a tendency to form it. Of these two, energy of activation is usually the controlling factor in deciding if a reaction will take place.
All right, so what determines a substance's energy of activation with air? (Wikipedia isn't helping.) In case it isn't obvious, I'm uneducated in chemistry, and an answer tailored accordingly, to the extent possible, would be ideal.