A field gun is an artillery piece.
Originally the term referred to smaller cannons that could be carried into combat by a moving field army, and moved about the field of battle. This was as opposed to siege cannon or mortars which were too large to be moved quickly, and would be used only in a prolonged siege.
Perhaps the most famous use of the field gun in terms of advanced tactics was Napoleon's use of very large wheels on the guns that allowed them to be moved quickly even during a battle. By moving the guns from point to point during the battle, enemy formations that were massing could be broken up to be handled by the infantry. The guns would then be moved to the next hot spot in the battle, dramtically increasing its firepower at any one point in time.
As the evolution of artillery continued, almost all guns of any size became capable of being moved at some speed. With few exceptions, even the largest siege weapons had become mobile by road or rail by the start of World War I, and evolution after that point tended to be towards smaller weapons with increased mobility. Although the Germans fielded a number of super-heavy guns (for no apparent good reason) in World War II, even these were rail or caterpiller-track mobile.
Thus since about the start of WWII the term has been applied to long-range artillery that fire at a low angle, as opposed to howitzers which tend to fire at higher angles. By the later stages of WWII almost all artillery was in the form of howizers of 105mm to 155mm, and the only common field gun of the era was the US 155mm Long Tom (a development of a French WWI weapon). The US Army tried the long-range gun again in the 1960s with a 175mm gun, but this was a failure, and after a rash of cracked barrels the gun was removed from service.
Today the gun finds itself in an area that seems to be gone for good. The class of small and highly mobile artillery has been filled with increasing capacity by the man-portable mortar, which replaced almost every artillery piece smaller than 105mm. Howitzers fill the middle ground, with the world rapidly standardizing on the 155mm NATO or 152mm fUSSR standards. The need for a long range weapon is filled by rocket artillery, or aircraft.
