The word rust is a common name for the specific compound Fe2O3, also known as iron oxide. The iron in this compound is iron(III), meaning it has lost three of its electrons when binding to the oxygen. Iron and alloys which contain iron are abundant materials that are found in cars, bikes, chains, and many other everyday objects around us, most of which are kept outside. The abundance of oxygen and water in the atmosphere readily reacts with the iron contained in many of these objects, and this is what causes many metallic outdoor objects to get rusty.
The overall reaction can be put simply as:

Within this overall scheme many smaller, more intricate reactions take place, one of which occurs when water comes into contact with iron. The iron dissolves in water and two important reactions take place, firstly the oxidation of iron from solid Fe to aqueous Fe(II), and secondly the reduction of water which proceeds using the four electrons generated in the previous reaction.

The products of these two reactions are then able to react with each other to form Fe(II) hydroxide, which will react with the oxygen formed in the previous equation to form Fe2O3, also known as rust.
Although the word rust refers to the oxidation of iron, other metals are still capable of corroding due to oxygen and water. An example of this type of corrosion occurs with copper, and one significant example stands famously on Liberty Island in New York Harbor. When it first arrived from France, the Statue of Liberty was brown in colour due to its copper containing coating.1 Over the years, this copper was oxidised by oxygen in the air to first form copper(I) oxide, which can then be oxidised further into copper(II) oxide.

Extensive coal burning caused elevated levels of CO2, SO3 and H2O to persist in the atmosphere, which then reacted with the CuO on the surface of the statue.1 The result of these reactions is three compounds, two of which are green and one which is blue. Over time the oxidation of copper has turned the Statue of Liberty completely green.
There are also metals that do not corrode but rather react with oxygen to form a protective layer on their surface. Aluminium is a very reactive metal and will readily react with oxygen in the air to form aluminium oxide. The surface of the aluminium eventually becomes coated with a thin layer of aluminium oxide which stops the layers underneath from also oxidising.
Depending on the metal, oxidation reactions with water and oxygen will have a number of different impacts on the material. The oxidation can be detrimental: forming rusted flakes through the oxidation of iron, change the colour of a structure: oxidation of copper surfaces causing a drastic colour change from brown to green, and even protect the metal from further damage: oxidation of surface aluminium atoms to form a protective aluminium oxide layer.
References
- R. A. Livingston, Environ. Sci. Technol., 1991, 25(8), 1400-1408
- S. E. Potts et al, J. Electrochem. Soc., 2011, 158(5), C132-C138