Many chemical reactions of the kind:
A + B --> C
exist that are exothermic -- they give out heat. Perhaps the simplest that one can write is
2H₂ + O₂ --> 2H₂O
(the formation of water from "burning" hydrogen).

This is a good example for fire, since it puts out a lot of heat !
The driving force for this reaction is that water is very stable, so it can be thought of as having a very low potential energy with respect to the reactants (H₂ and O₂). The heat released is a measure of the change in potential energy.

However, we usually associate fire with light as well, and the flame formed by the burning of hydrogen is often colorless. We will return to the issue of the color of flames presently.

Oxygen is not essential for fire. For example, when a small piece of sodium is heated and lowered into a jar of chlorine gas, it burns with a bright yellow flame. The product is sodium chloride (common salt)and once again, the reaction is driven strongly (with lots of heat being released) because common salt is so stable.

So what makes some flames colored (such as the flame from a candle)and some colorless (hydrogen)? The color of a hot body (the gases associated with a flame) can arise due to something called black-body radiation. Any hot body glows, emitting light over a very wide range of color and the color of the body is usually an effective indicator of the temperature. This is the principle of a normal light bulb where the filament runs at around 2,500C and the light is nearly white, though a bit yellow. If you run the bulb from a dimmer you can see that it changes to red as it cools down.

In addition, color can arise from electronic excitations. The blue color of gas burning in a kitchen range is a good example of this. The carbon-containing molecules of the gas are broken up, and for a very short time they form high-energy (“excited”) C₂ molecules, which lose their energy by emitting blue light. Because there is plenty of air mixed in by the burner these are quickly burnt up, so the only color is this blue.

With a candle flame things are more complicated, The only way air can get to the flame is from the outside, so at that outside edge we can see the same blue color from excited C₂ molecules. However, not enough air can get inside the flame, so the C₂ molecules start combining with each other to form very small, very hot particles of carbon (soot) which emit black body radiation at a temperature of about 1100C, cooler than the light bulb so it is yellow.

In addition to the emission from molecules mentioned above, color can arise from excited atoms. A good example is fireworks, where the color is no longer an indicator of the temperature. Salts of the alkali and alkaline earth metals (Li, Na, K, Ba etc.) are usually added to a brightly burning fuel (such as Mg or Al powder). When heated to high temperatures, electronic processes in these alkali and alkaline earth atoms yield characteristic colors. The yellow flame from sodium burning in chlorine, or the yellow color from a sodium lamp on the highway, are all the same emission from sodium atoms. These electronic processes involve electrons in the atoms jumping between levels, and when they jump from a level with higher energy to one with lower energy, they release the difference as light.

Many thanks to Prof. Lloyd for their contributions to Answer #1