What is the difference between carbon steel, mild steel and stainless steel ?

Mild steel contains 0.16–0.29% carbon.
Carbon steel has a carbon content in the range of 0.30–1.70% by weight.
Stainless steel has a minimum of 11% chromium content by mass.

Only stainless steel does not corrode.

There are lots of stainless steels. Steel, by definition is an alloy containing carbon. The different alloyed elements give various steels their properties -- including strength, stiffness, brittleness, corrosion resistance, among others.

"Stainless Steels" don't corrode because one of their alloying metals (usually chromium or molybdenum) forms a passive migrates to the surface of the solid and forms a thin, hard oxide layer that is difficult to get through. In addition, there are electrochemical reasons chromium and some other elements are resistant to corrosion.