Cyclooxygenase (COX)  catalyzes the conversion of the free essential fatty acids to prostanoids by a two-step process.  
In the first step, two molecules of O are added as two peroxide linkages and a 5-member carbon ring is forged near the middle of the fatty acid chain.  This forms the short-lived, unstable intermediate Prostaglandin G (PGG).  
One of the peroxide linkages sheds a single oxygen, forming PGH.  (See diagrams and more detail at Cyclooxygenase).  
All other prostanoids originate from PGH (as PGH, PGH, or PGH). 
The image at right shows how PGH (derived from Arachidonic acid)  is converted:
* By PGE synthetase into PGE2 (which in turn is converted into PGF2)
* By PGD synthetase into PGD2
* By Prostacyclin synthase into prostacyclin (PGI2)
* By Thromboxane synthase into thromboxanes TXA 
Arachidonic acid is made up of a 20-Carbon unnatural poly unsaturated Omega-fatty acid. Arachidonic acid presents within the phospholipid bi-layer as well as in the plasma membrane of a cell. With Arachidonic acid prostaglandins are formed through synthesis and oxygenation of enzymes. Active lipids in the oxylipin family derive from the synthesis of Cyclooxygenase or Prostaglandins.  
The three classes of prostanoids have distinctive rings in the center of the molecule.  They differ in their structures and do not share common structure as Thromboxane. The PGH compounds (parents to all the rest) have a 5-carbon ring, bridged by two oxygens (a peroxide.)   The derived prostaglandins contain a single, unsaturated 5-carbon ring.  In prostacyclins, this ring is conjoined to another oxygen-containing ring.  In thromboxanes the ring becomes a 6-member ring with one oxygen.
Production of PGE in bacterial and viral infections appear to be stimulated by certain cytokines, e.g., interleukin-1.