A chemical reaction is what happens when you introduce two different compounds together and get different compounds from the mixture.
For a simplified example, rust is what happens when iron is mixed with oxygen and water. The hydrogen in the water can combine to make acids, which loosen the iron particles. The iron then combines more easily with the oxygen, creating iron oxide, or what we know as "rust."
Another example is the "baking soda volcano." Baking soda is a base while vinegar is an acid. They combine to make carbonic acid, and that acid almost immediately breaks down to form water and carbon dioxide. The gas release from the water is what causes the "eruption."