There are three types of other products, substitute products, joint products, and independent products.  If two products are independent then a change in the cost of production of one product does not affect the cost of production of the other product.  An example of independent products would be wheat and computers.

A substitute product is a product that can be produced with the same (or similar) set of inputs.  When two goods are substitute products an increase in the price of one product increases the production cost of the other product.  This is because they use the same (or similar) set of inputs.  An example of substitute products would be gasoline and plastics.  Both gasoline and plastics use oil as the base material in their production.  As the price of plastics increases more firms will begin producing plastics (Recall: an increase in price increases output), which increases the demand for oil.  The increase in the demand for oil causes the price of oil to increase which causes an increase in the cost of production for all goods that use oil, including gasoline.  So an increase in the price of a substitute product causes the cost of production to increase and supply to decrease.

Joint products are goods that are produced together because it is difficult or absurd to produce them separately.  When two goods are joint products an increase in the price of one good causes the supply of the other good to increase.  An example of joint products would be beef and leather.  An increase in the price of beef would cause more beef to be produced.  Because you can not produce beef without producing leather (at least raw leather) the increased production of beef would cause an increase in the production of leather.