The particle structure of a burning flame is actually a subject on which not much is published. For the purposes of simplicity, we'll only consider some of the simplest flames in our discussion. When you burn hydrogen and oxygen, what you are really doing is causing a chemical change, similar to rusting or old newspapers turning brown. What separates the flame however, is the speed and violence of the reaction. What is actually happening in the flame is that the individual atoms are shedding some or all of their electrons, and recombining into molecules. The only way for the equation to balance out, however, is through the release of energy.

When hydrogen and oxygen combine, the resulting product, H20 is more stable together as a molecule than the previous two components were when they were individuals. Previously they had a higher energy state. Now, once they've combined, their total "energy state" is lower. Where does that extra energy go? Well, it goes into the heat and light that you see coming from the flame.

Now - to what's really going on in there, which is the heart of the question. Oxygen normally floats around as a pair, O2. Hydrogen can be found as a pair, but is usually found alone, H. The burning process first absorbs energy to split the O's apart (the spark, etc) and then those two free O's almost instantaneously find two hydrogens each to pair up with, and bond with them by attracting their single electrons to fill the two "holes" in the O shell, releasing raw energy in the process. Sometimes, it takes longer to find the second H. Therefore, within the flame, you have in differing amounts - O2, H, O, OH, H2O, heat, light, and the occasional free electron or proton that has dissociated from its parent atom. That is the particle structure of a flame. It is a mixture of the starting components, the ending components, and the intermediate products.

This whole thing gets way more complex when you consider something like ethanol or gasoline or acetone, etc. Then you end up with all kind of intermediate combinations of things, some of which may not last all the way through until burning in complete.