Corrosion and rusting are two chemical processes, which result in disintegration of materials.
Corrosion
When a material reacts with the external environment, over a time, its structure will be deteriorated, and breaks down into small pieces. Ultimately, it can disintegrate into the atomic level. This is known as corrosion. Most commonly this happens to metals. When exposed to the external environment, metals will undergo oxidation reactions with oxygen in the atmosphere. Other than metals, materials like polymers, ceramics can also undergo disintegration.
Rusting.
Corrosion of iron or iron containing materials is called rusting. Rusting is chemically oxidation of iron to oxides of iron or hydroxides in presence of oxygen and water.