Nuclear thermal rockets can be categorized by the construction of their reactor. The most typical type uses a conventional (albiet light-weight) reactor running at high temperatures to heat the working fluid that is moving through the reactor core. This is known as the solid-core design, and is the simplest design to construct.
The solid-core design has the downside that it can only be run at temperatures below the melting point of the materials used in the reactor core. Since the effeciency of a rocket engine is strongly related to the temperature of the working fluid, the solid-core design needs to be constructed of materials that remain strong at as high a temperature as possible. However even the most advanced materials melt at temperatures below that which the fuel can actually create, meaning that much of the potential energy of the reactions is lost.
In order to increase the temperature it is possible to mix the nuclear fuel into the working fluid, and allow the reaction to take place in the mixture itself. This is the so-called liquid-core engine, which can operate at higher temperatures beyond the melting point of the fuel. In this case the maximum temperature is whatever the container wall can handle, which is typically a neutron reflector of some sort.
However these engines are difficult to build; the reaction time of the nuclear fuel is much higher than the heating time of the working fluid, meaning that some system must be used to trap the fuel inside the engine while still allowing the working fluid to easily exit through the nozzle. Most liquid-phase engines have focussed on rotating the fuel/fluid mixure at very high speeds, forcing the fuel to the outside due to centrifugual force (uranium is heavier than hydrogen).
The final classification is, obviously, the gas-core engine. This is a modification to the liquid-core design which uses rapid circulation of the fluid to create a toroidal pocket of gasseous uranium fuel in the middle of the reactor, surrounded by hydrogen. In this case the fuel does not touch the reactor wall at all, so temperatures could reach several tens of thousands of degrees, which would allow specific impulses of 3000 to 5000 seconds.
Although engineering studies of all of these designs were made, only the solid-core engine was ever built. Development of such engines started under the aegis of the Atomic Energy Commission in 1956 as Project Rover, with work on a suitable reactor starting at LANL. Two basic designs came from this project, Kiwi and NRX.
Kiwi was the first to be fired, starting in July 1959 with Kiwi 1. This was unlike later tests because the engine design could not really be used, the core was simply a stack of uncoated uranium oxide plates onto which the hydrogen was dumped. Nevertheless it generated 70 MW and produced an exhaust of 2683 K. Two additional tests of the basic concept, A' and A3, added coatings to the plates to test fuel rod concepts.
The Kiwi B series fully developed the fuel system, which consisted of the uranium fuel in the form of tiny UO2 spheres embedded in a low-boron graphite matrix, and then coated with niobium carbide. Nineteen holes ran the length of the bundles, and through these holes the liquid hydrogen flowed for cooling. A final change introduced during the Kiwi program changed the fuel to uranium carbide, which was run for the last time in 1964.
Using information developed from the Kiwi series, the Phoebus series developed much larger reactors. The first 1A test in June 1965 ran for over 10 minutes at 1090 MW, with an exhaust temperature of 2370 K. The B run in February 1967 improved this to 1500 MW for 30 minutes. The final 2A test in June 1968 ran for over 12 minutes at 4,000 MW, the most powerful nuclear reactor ever built. For contrast, the largest hydroelectric plant in the world, Itaipu, produces 12,600 MW, 25% of all the power used in Brazil.
A smaller version of Kiwi, the Peeweee was also built. It was fired serveral times at 500MW in order to test coatings made of zirconium carbide (instead of niobium carbide) but also increased the power density of the system. An unrelated water-cooled system known as NF-1 (for Nuclear Furnace) was used for future materials testing.
While Kiwi was being run, NASA joined the effort with their NERVA program (Nuclear Engine for Rocket Vehicle Applications). Unlike the AEC work which was intended to study the reactor design itself, NERVA was aiming to produce a real engine that could be deployed on space missions. A 75,000lbs thrust baseline design was considered for some time as the upper stages for the Saturn V, in place of the J-2s[?] that were actually flown.
The design that eventually developed, known as NRX, started testing in September 1964. The final engine in this series was the XE, which was the first designed to be fired in a downward position (like a "real" rocket engine) and was fired twenty-eight times in March 1968. The series all generated 1100 MWt, and many of the tests concluded only when the test-stand ran out of hydrogen fuel. EX produced the baseline 75,000 lbs thrust that NERVA required.
All of these designs also shared a number of problems that were never completely cured. The engines were also quite easy to break, and on many firings the vibrations inside the reactors cracked the fuel bundles and caused the reactors to break apart. This problem was largely solved by the end of the program, and related work at Argonne National Laboratory looked like it could produce a much stronger fuel bundle. However, while the graphite construction was indeed able to be heated to high temperatures, it likewise eroded quite heavily due to the hydrogen. The coatings never wholly solved this problem, and significant "losses" of fuel occurred on most firings. This problem did not look like it would be solved any time soon.
The NERVA/Rover project was eventually cancelled in 1972 with the general wind-down of NASA in the post-Apollo era. It was also clear that there would be intense public outcry against any attempt to use a nuclear engine.
A related design that saw some work, but never made it to the prototype stage, was Dumbo. Dumbo was similar to Kiwi/NRX in concept, but used more advanced construction techniques to lower the weight of the reactor. The Dumbo reactor consisted of several large tubes (more like barrels) which were in turn constructed of stacked plates of corregated material. The corregations were lined up so that the resulting stack had channels running from the inside to the outside. Some of these channels were filled with fuel, others with a moderator, and some were left open. Fuel was pumped into the middle of the tube, and would be heated by the fuel as it traveled through the channels as it worked its way to the outside. The resulting system was lighter for any particular amount of fuel. The project developed some initial reactor designs, and appeared to be feasible