What makes nails rust?

To understand why nails rust is necessary to start from the metal of which nails are made: Iron.

Iron is a metal that when in contact with water corrodes, and the result of the corrosion process of iron is the formation of Rust.

What is rust? Rust is a generic term to describe a series of different oxides, Fe(OH)₂, Fe(OH)₃, FeO(OH), Fe₂O₃.H₂O that forms when iron corrodes.

Fe₂O₃ is a red oxide called Hematite and is reponsible for the typical color of a rusted object.

Now you would like to know why water makes iron rust?

Water corrodes iron because the oxygen in the air dissolves in the water and reacts with iron. There are always two distinct chemical reactions in a corrosion process, the basic chemistry of corrosion is:

1) Anodic Dissolution of Metal (Iron) that goes into solution (water)

Fe  ----->   Fe²⁺  +   2e-

2) Cathodic Reduction of Oxygen dissolved in water

O₂  + 2H₂O + 4e^- ---->  4OH^-  

The final reaction is: 

Fe²⁺ + 2OH^-  ----->  Fe(OH)₂

This oxide will then further reacts with oxygen to give the final red product:

Fe₂O₃.H₂O