The escort fighter was a World War II concept for a fighter aircraft designed to escort bombers to and from their targets. An escort fighter needed range long enough to reach the target, loiter over it for the duration of the raid to defend the bombers, and return.
A number of twin-engined heavy fighters with high fuel capacity were designed for escort duties before World War II. Such heavy fighters largely failed in their intended escort role during the war, as they were outmaneuvered by more agile single-engined fighters. As the war progressed, longer-range fighter designs and the use of drop tanks allowed single-engined fighters to perform escort duties.
In the post-war era the introduction of jet engines and their inherent short range made escort fighters very difficult to build. The related concept of a penetration fighter emerged briefly in the 1950s and again in the 1960s, but did not result in any production aircraft. While the importance of dedicated escort fighter aircraft has waned following the end of the Second World War, the escort role has been diminished since modern air combat doctrine places a heavy emphasis on the idea of Air Superiority, and its importance in the ability of an air force to carry out effective operations. Air superiority is defined as a situation in which an air force dominates an airspace to such a degree as to be able to carry out any operations with no interference from enemy air combatants. Fighting an opponent with air superiority in a given battlespace is much harder, as any offensive or defensive tactics are likely to be overwhelmed. 
During the Battle of Britain, the Luftwaffe used Messerschmitt Bf 109s and Bf 110s based in France as escort fighter-bombers. Although flying from relatively close airfields in France, the Bf 109 was operating at the extreme of its range and unable to remain for long with the bombers if it was to have fuel to return, while the Bf 110, specifically designed for the escort role, had inferior performance and was easily outperformed by the Royal Air Force Supermarine Spitfires and Hawker Hurricanes. Flying with the Heinkel He 111 and Junkers Ju 88 medium bombers, they would drop their light bomb-loads and--for a brief period--fight off the British fighters.
The U.S. Army Air Forces' precision strategic bombing campaign against German industries was only possible during the day. At first, this was not seen as an issue; the Forces' Boeing B-17 Flying Fortress and Consolidated B-24 Liberator bombers were the most heavily armed aircraft of the time. Close formations of them were planned, creating a crossfire of .50 caliber machine-guns that would fend off the enemy with no need for a fighter escort.
In the early stages of World War II, bombing runs often took place without escort fighters. German fighter pilots were scrambled to deal with these raids, and soon learned that it was much easier for them to take out formations which were unescorted as opposed to those who were escorted. As a result, fighters would attack bomber formations which were on long range operations, as they would not have escorts with them. USAAF bomber losses gradually increased, and experimental "gunships" like the YB-40 did nothing to reduce them. This prompted the United States Army Air Force to come to the conclusion that their B-17 bombers needed to do something to reduce losses, and eventually led to the development of long range escort fighters. 
It was not until the introduction of the Lockheed P-38 Lightning and Republic P-47 Thunderbolt fighters that the bombing raids could claim a measure of success. Able to carry large Lockheed-designed drop tanks, the fighters were able to escort the bombers for much of their missions. The first Allied fighters over Berlin were 55th Fighter Group P-38s on March 3, 1944. When the Merlin-powered North American P-51 Mustang was introduced, with a laminar-flow wing for efficiency, the final escort fighter development of the war was complete.
The successes of the P-47N and P-51 gave the impression that the escort fighter was a concept worth continuing after the end of the war. The high fuel use of early jet engines made such aircraft difficult to design, and a number of experimental designs were tried that used mixed power, typically a turboprop and jet, but these failed to meet performance requirements. A new concept, the XF-85 Goblin microfighter, planned to act as a parasite fighter for the Convair B-36, was tested with a B-29 Superfortress and found to be utterly impossible to use operationally. Later the FICON project attempted a similar solution, docking jet fighters with heavy bombers via a trapeze mechanism or their wingtips.
Whilst projects for dedicated escort fighters such as the XF-85 Goblin came to nothing, the advancement of technology and the nature of warfare of the wars being fought allowed the role of fighter escort to gradually merge with fighter types, so the term fell out of use. In Korea, the F-80 Shooting Star and later the F-86 Sabre escorted B-29 heavy bombers and F-84 Thunderjet strike fighters.
Although the XB-70 Valkyrie, North American Aviation's Mach 3 bomber, was intended to be immune to enemy attack due to its speed, North American Aviation very briefly proposed the XF-108 Rapier interceptor for the escort role. In this case, the term "penetration fighter" was used, as the aircraft was not expected to actually escort the bombers, and was instead intended to fly into Soviet airspace well in advance of the bombers and attack the Soviet interceptors long before they could approach the bombers.
With the development of guided missiles, particularly surface-to-air missiles, plans for dedicated escort fighters designed to escort nuclear bombers gradually faded from the scene. Missile technology meant that interceptors would rarely engage bombers directly, if ever, and the escorts could do little against missiles. At the same time, the advancement of land and submarine-based ballistic missiles relegated bombers to a lower importance -- they became just a single element of the nuclear triad in the US, and largely ignored entirely in the USSR. Furthermore, with the concept of mutually assured destruction high on the political agenda throughout the Cold War, a nuclear exchange became ever less likely, leaving existing fighter designs more than adequate for their protection in the wars being fought. In Vietnam for instance, F-4 Phantom IIs and sometimes F-8 Crusaders escorted the American bombers such as B-52 Stratofortresses, F-105 Thunderchiefs and A-4 Skyhawks. In some cases the missions of F-4 were "mixed", when some F-4 were equipped with bombs, and some F-4 acted as escorts (similar cases occurred with F-8).
The advent of the air superiority fighter, such as the F-15 Eagle, meant that high value assets like tankers, AEW&C, command platforms, bombers and attack aircraft would be protected by air superiority fighters, sometimes flying far afield and ahead of them, engaging distant enemy air units, rather than by direct escorts staying in sight nearby.
The US Air Force is currently developing a new fighter escort for the anticipated next generation B-21.