Fire is caused by the oxidation of molecular bonds and in some materials like wood the energy released by oxidation is sufficient to create a runaway chain reaction called combustion
The flame of a fire is a more interesting question to think about. We know blue flames are hotter than red flames. This is because on the Electomagnetic spectrum, the blue photons have a greater energy than the red photons. When we burned the wood, we oxidized its bonds and its electrons became excited, when these electrons fall back down to their normal dexcited state, they emit a photon. This photon has a certain energy that is equal to the size of the electron's 'fall', so the bigger the fall, the more energetic the photon becomes.
Heat is another form of 'light' or more correctly speaking Electromagnetic Radiation. It is light that is slightly out of our range of vision, just below the red end of the visual spectrum. If you don't believe me, turn on your stove grill and notice that when it reaches a certain heat, it starts glowing red. If stoves could get even hotter, the grill would start looking bluish. Heat, also called Infrared Radiation, is produced when atoms or molecules jitter sideways back and forth in an excited vibratory manner.
Electrons can orbit individual atoms, or they can orbit molecules. If they are bombarded with radiation energy (like they are when you light a piece of wood on fire) the electrons jump up and fall down, jump up and fall down. Some electrons even escape their orbits in a process called ionization. A collection of these ionized electrons is called a plasma. The flame of a lighter is a plasma of these ionized electrons.
One metaphor for how fire is sustained by a chain reaction goes like this...
Imagine you are standing in the center of a room made of steel. You have a lighter in your hand. When you flick your lighter on it shoots photons outwards in every possible direction. Think of these photons as bullets that hit the steel walls, break into fragments and richochet all around the room. If we turn the lighter off now, we'd expect that since there are no more bullets being fired at the walls, that the richochets would die down and lose energy and the room would become silent. But, if the walls, when struck by a bullet, give up some of the energy holding them together, the ricochets continue on and on until there is no wall left. Without fuel there can be no fire.
