A flame is just the visible part of combustion. In most cases we encounter, combustion is the oxidation of carbon ( in gasoline, natural gas, or wood) to carbon dioxide. (Oxidize means adding oxygen, and here we add 2 oxygen atoms to carbon.) More specifically, the color of the flame comes from exciting atoms or molecules with the heat generated by the combustion. As you know, fire is very hot, and this energy is enough to excite atoms in the fire and make them emit light. The yellow colour in a flame generally comes from carbon.

You may be familiar with the fact that matter can come in three different phases: first, a substance starts off as a solid (for example, ice), but as you heat it, it melts into a liquid (like water).If you keep heating the liquid, it evaporates and becomes a gas (like water vapor). But what happens if you keep heating the gas? Well, the temperature of some matter basically measures how much the atoms and molecules that make it up are moving around, so if you take a gas and start heating it more and more, the atoms and molecules crash into each other harder and harder, until they break apart, and the electrons are removed from the atoms they were orbiting. This process is called ionization, and the charged particles that are left over are called ions. If you have a whole bunch of ions, then you get what's called a plasma, which is sort of like a fourth phase of matter.

Now, when you burn something, the chemical reactions give off gases and heat; as the gases are heated, they turn into a plasma, which is exactly what a flame is. So there it is: a flame is a plasma made up of the ionized gases created during the burning.

A flame is hot gas (normally air) that is so hot that it glows. The gas is hot because of a chemical reaction, usually involving oxygen reacting with other stuff, that produces the heat. Flame is also more of a phenomenon than a thing: if you were to track a single molecule of gas, it would get sucked into the flame cold, get heated up and become part of the flame, and then float out the top, cool off, and stop glowing (and thus no longer be part of the flame).