Most rockets use chemical engines which combine a fuel and an oxidizer to accelerate the exhaust products and provide thrust. Isps of chemical rockets max out at about 450 seconds. Nuclear thermal rockets get Isp from 900 seconds on up. This is because their exhaust velocity is not limited by the energy of a chemical reaction, as they use the heat from a nuclear reaction to accelerate hydrogen by expanding it inside the core of a reactor.
This is the simplest type of nuclear thermal propulsion, and NASA did some tests on it in the '50s and '60s. The problem is that it's performance is limited by the melting point of the reactor. There are a couple of types of nuclear thermal propulsion that avoid this problem in various ways:
Twisted ribbon, pebble bed
By changing the shape of the fuel rods in the reactor, you can increase the heat transfer to the propellant. Twisted ribbon looks like this:
Pebble bed reactors are made out of "pebbles" of fuel that look like this;
Pulsed solid-core
This type of engine works by pulsing the power at which the reactor runs at from its normal power level to far beyond what the materials that it is made could withstand. As the temperatures are momentary, the cooling system can keep it from melting down. This is even better than it sounds, as neutrons generated while it is at maximum power will heat the propellant even further.
Liquid core
Solving the problem of preventing your reactor from melting down by designing it to run while molten! Outside-the-box thinking.
Problem include keeping your molten core from escaping with the propellent.
Isp 1,500 seconds.
Gas core
The logical extension of the liquid core design by allowing the uranium to turn not just liquid but to gas.
The problem of preventing the uranium plasma from escaping is exacerbated with gas core.
Isps 1,800 seconds to 7,000 seconds with advanced designs.
Nuclear lightbulb
It works like an easybake oven, except the lightbulb is a uranium plasma contained by a quartz vessel that doesn't melt because quartz is transparent to the majority of the radiation that is emitted, and the cake is hydrogen travelling at 20 km/s. Ages 10 and up.
Isps between 2,000 and 3,000 seconds.
Fission fragment
This skips a separate propellant entirely and uses the split atoms from the fission as propellant. The original concept used discs coated with the fuel, but a more efficient design uses ground-up fuel (around 100 nanometers) which are magnetically contained before they fission.
Isp 100,000 to 1,000,000 seconds.
Nuclear pulse
Uses nuclear bombs as propulsion, by dropping them behind a strong plate that dampens the shock and heat enough that anyone on board doesn't die. It apparently would work, and was even tested with conventional explosives. Development stopped mostly due to the nuclear test ban treaty and that this is best suited to lift spacecraft from the ground to orbit.[This] is not nuts, [this] is super-nuts-Richard Courant on viewing an nuclear pulse engine test
Isp 4,000 to 7,000 (with fusion bombs)
Nuclear salt water
This is one of the most crazy concepts for a rocket engine ever made. Its fuel is 20% enriched uranium mixed at 2% with water. It is stored in tubes coated with some kind of moderator which prevents it from reaching critical mass inside the tanks. When it is injected into the combustion chamber, the uranium reaches critical mass and begins a nuclear detonation. This can be sustained with more of the fuel being injected into the chamber. This has the advantage of the nuclear pulse engine's high efficiency, with continuous thrust and the ability to work at small scales.
