Robert Frost once wrote a poem about coming across an old woodpile in the forest. In describing it he refers to the "slow, smokeless burning of decay."
What a great line of poetry. Neat part about it is, it's also technically correct. Decay is an extremely slow burning process. Or, you could say, fire is an extremely fast decaying process. In either case, what you have is the combination of carbon, oxygen, and hydrogen.
Carbon and oxygen atoms love to link up with each other and form carbon dioxide. Right behind them in the race to merge come hydrogen and oxygen, which love to form di-hydrogen oxide. It's this kind of rapid merging of atoms that we call fire.
Hold on! you say. Fire has, like, flames and stuff!
That's because when carbon and oxygen merge to form carbon dioxide, for example, they give off energy. If the merging process is very slow, say in a decaying woodpile, that energy will be very subtle. Speed it up a bit, though, and the energy can be felt as heat. Even more and you can see it as light. That light is what we know as fire. Fire is simply the light and heat coming from the rapid combination of atoms.
Now, all living things are chock full of carbon and hydrogen, and the air has plenty of oxygen in it. That being said, when a pile of dead wood is left out for a while, the carbon and hydrogen atoms slowly combine with oxygen, leaving behind a gradually diminished pile. There are also microorganisms helping things along, but even without them, there would still be the slow, smokeless burning of decay.