Definition
Think of the flame on a kitchen gas stove. When a material reacts fiercely with oxygen and gives off heat and light, that is a combustion reaction. Everything we casually call 'burning' is what science calls combustion.
The Three Things Every Fire Needs
Picture lighting a campfire. To get a fire going, you need something that can burn (fuel, like wood). You need oxygen from the air (the air you fan in). And you need a spark or flame to raise the temperature (a temperature at or above the ignition point). If even one of these three is missing, the fire will never catch.
These three are called the fire triangle. The ignition point (the lowest temperature at which something catches fire on its own) is different for every material. Paper ignites at only about 230°C, but a thick log needs far more heat before it starts to burn.
When firefighters spray water, they are lowering the temperature below the ignition point. When they throw a blanket over a fire, they are cutting off the oxygen. So a combustion reaction is a special kind of bonding that starts only when all three conditions line up just right.
What Happens to Atoms Inside a Flame?
Among common fuels, charcoal is mostly carbon, and the natural gas piped into homes is made of carbon and hydrogen. When the fuel meets oxygen and burns, the bonds between its original atoms break, and the atoms pair up with oxygen in new ways. Carbon joins with oxygen to become carbon dioxide. Hydrogen joins with oxygen to become water (water vapor).
When there is plenty of oxygen, the fuel burns completely and leaves only carbon dioxide and water. This is called 'complete combustion.' It makes almost no soot and gives a hot blue flame. The reason a gas stove flame is blue is that air is flowing in freely.
When oxygen is short, it's the opposite. Carbon meets less oxygen and turns into carbon monoxide, or unburned bits of carbon come out as soot (black smoke). This is called 'incomplete combustion.' Carbon monoxide has no smell, but it is a dangerous poisonous gas that keeps your body from carrying oxygen.
A Little More Precisely
To be a little more precise, a combustion reaction is one kind of 'redox reaction,' in which electrons are passed from one substance to another. Iron slowly rusting in the air is also a reaction of iron combining with oxygen, so chemically it works on the same principle as combustion.
The difference is how fast the reaction happens. Rusting is so slow that you can hardly feel any heat. Combustion pours out a huge amount of energy in the blink of an eye, so the bright light and intense heat show on the outside.
The 'cellular respiration' that lets the cells in your body burn food (glucose) to keep you warm and give you strength is a very slow, finely controlled cousin of combustion. In the end, combustion is a law of nature: a material that holds energy meets oxygen, turns into a more stable substance, and releases that energy.
Common misconceptions
When something burns and only ash is left, mass has disappeared and it has become lighter.
The material didn't vanish. It turned into gases like carbon dioxide and water vapor and drifted off into the air. If you collected all of that gas and weighed it too, the total mass would be exactly the same as before the reaction.
Where you meet it
When you turn on a gas stove, the natural gas (methane) reacts with oxygen and shoots out a blue flame.
As wood burns in a campfire, it gives off warm heat and an orange glow.
Inside a car engine, gasoline burns explosively and creates the force that turns the wheels.
In one sentence
A combustion reaction is an oxidation reaction that gives off light and heat all at once when fuel, oxygen, and enough temperature come together.