Oxidation is the process of an element losing electrons.  For example, iron will transfer two of its electrons to oxygen, forming an oxide.  This occurs all throughout as an unintended part of the steelmaking process.
Oxygen blowing is a method of steelmaking where oxygen is blown through pig iron to lower the carbon content.  Oxygen forms oxides with the unwanted elements, such as carbon, silicon, phosphorus, and manganese, which appear from various stages of the manufacturing process.  These oxides will float to the top of the steel pool and remove themselves from the pig iron.  However, some of the oxygen will also react with the iron itself.
Due to the high temperatures involved in smelting, oxygen in the air may dissolve into the molten iron while it is being poured.  Slag, a byproduct left over after the smelting process, is used to further absorb impurities such as sulfur or oxides and protect steel from further oxidation. However, it can still be responsible for some oxidation.
Some processes, while still able to lead to oxidation, are not relevant to the oxygen content of steel during its manufacture.  For example, rust is a red iron oxide that forms when the iron in steel reacts with the oxygen or water in the air.  This usually only occurs once the steel has been in use for varying lengths of time.  Some physical components of the steelmaking process itself, such as the electric arc furnace, may also wear down and oxidize.  This problem is typically dealt with by the use of refractory metals, which resist environmental conditions.
If steel is not properly deoxidized, it will have lost various properties such as tensile strength, ductility, toughness, weldability, polishability, and machinability.  This is due to forming non-metallic inclusions and gas pores, bubbles of gas that get trapped during the solidification process of steel.