A blitzkrieg is an operational level military doctrine which calls for a quick surprise attack by massive ground and air forces. "Blitzkrieg" is from the German words for lighting ("blitz") and war ("krieg"). The term was not used by German military planners, and had its origins at the Propaganda Ministry of Josef Goebbels. It was used in press accounts to describe the lightning speed of the German military during World War II. Modern American "Shock and Awe" doctrine is based upon the lessons of blitzkrieg.
Before World War II
Colonel John Fuller, the chief of staff of the British Tank Corps, is credited with developing this tactic after his extreme disappointment with the effect of tanks during World War I. His ideas, which were ignored by the British Army, called for long-range mass tank attacks. He envisioned this attack to be supported by air power as well as motorized infantry and artillery support. He wrote two books in the Twenties which detailed his ideas: Reformation of War and Foundation of the Science of War. In 1926, the German Army began ordering tanks and vehicles which would allow them to implement Colonel Fuller's ideas. 
In 1937 Heinz Guderian wrote the book Achtung - Panzer! describing the combined arms tactics used initially by the Germans in WWII and later on by armies throughout the world. The Germans tested an application of these tactics in 1938 during the Spanish Civil War and as a result German High Command formulated plans to dismantle Polish defenses in 1939. 
During World War II
Blitzkrieg was a maneuver tactic where armored columns would break through the enemy lines, supported by artillery and attack aircraft. Once the columns were through the line, they would not stop but continue into the enemy rear, disrupting their supply lines and cutting off units. The armored columns were followed by infantry units that would exploit the gaps and surround the isolated enemy units. The tactic worked most effectively in Europe against the French and the Netherlands, and in the beginning of the conflict against the Soviet Union. It's success was due to a combination of surprise and favorable terrain.
There was three flaws to the Blitzkrieg tactic. First, it was heavily dependent on good weather and good terrain. If the terrain was hilly, combat was in an urban area, or if the weather turned the ground muddy (as in the Soviet Union), the armored units would slow down and become vulnerable to anti-tank units.
Second, air support was vital. Without it, the tanks were vulnerable to counter-attacks from enemy aircraft. As the war progressed and the Germans lost control of the air, their panzer units suffered heavy losses from Allied air-to-ground aircraft such as the Soviet IL-2 and American P-47 Thunderbolt. German success in the Battle of the Bulge depended in part on the existence of cloud cover that would keep Allied air forces grounded.
Third, the blitzkrieg could be broken up by channeling the attack (holding the flanks) and attacking the point where the infantry and armored units met. This would separate the two and expose the armor to counter-attacks. The French and British used this tactic at the Battle of Arras in 1940, but lacked the armored units required to exploit their limited success.
The Soviets used all the above and had the additional fortune of vast swaths of land to design a defense-in-depth. As the panzer units moved farther and farther from their supplies, gaps would appear in their columns. These could then be attacked to break up the armor. Also, the Soviets used minefields to further disrupt the German armor.
By 1944, blitzkrieg as a German tactic was finished. The last actual appearance of blitzkrieg in the war was when General George S. Patton's Third Army broke out of the Normandy Pocket in Operation Cobra. The Allied columns covered 600 miles in just under two weeks.
Blitzkrieg after the war
The first significant use of blitzkrieg tactics in the post-war era was in the 1967 Six Day War when Israel, after destroying almost the entire Egyptian air force on the ground in a little more than an hour, sent a three pronged tank attack south to seize the Sinai peninsula and north to occupy the Golan heights.
The ground phase of the first Gulf War, lasting only four days, followed a blitzkrieg-style attack with US, British, and French tanks that poured deep into southern Iraq in what General Norman Schwarzkopf termed a "Hail Mary" pass before hooking right and cutting off the escaping Iraqi forces, which were then demolished in detail by tanks and aircraft.
The resumption of hostilities in Iraq that led to the Iraq War relied upon "Shock and Awe" a derivative of blitzkrieg doctrine, but also a tactic used since the American Civil War, when massed artillery bombardment was first used to weaken and demoralize enemy forces before an advance.
Frieser, K.H. 'The Blitzkrieg Legend: The 1940 Campaign in the West'