LANTR stands for LOX-Augmented Nuclear Thermal Rocket. It is a bi-mode spacecraft rocket motor design intended for high-speed expeditions within the Solar System. Nuclear rockets are attractive options for interplanetary travel because they have much higher specific impulse than pure chemical engines like the SSME. However, they are limited in that their maximum practical output (thrust) is much lower than chemical engines. Think of it this way: the nuclear engine is highly efficient (Toyota Prius) but the chemical engine is more powerful (V8 truck). Now think of getting to Mars, say, this way: It's a cross-the-USA roadtrip with no ability to stop for gas, but in order to start out you have to tow a huge trailer up a really steep driveway.
The driveway, of course, is a gravity well. While the truck is good at towing trailers up hills, you can't stuff enough fuel in the truck to make it across the US no matter what you do. While you can put extra tanks in the Prius and make it to, say, California from Florida, there's no way the Prius can tow a two-ton trailer up that driveway. The trailer, of course, contains all the stuff that your astronauts need to survive on the trip, on Mars, and on the way home (food, oxygen, equipment, etc.)
The LANTR attempts to address this problem. It is a nuclear thermal rocket - in other words, it heats a fuel, typically hydrogen, up to very high temperatures and tosses it out the rear, creating thrust. Since the energy used in this process all comes from the nuclear fuel, it's very very efficient. But this engine doesn't have enough oomph to, say, launch from Mars back into space so that the astronauts can return home. In a LANTR, however, you give up some of your fuel space for an oxidizer - in this case, LOX (liquid oxygen). Then, when you need a short-term boost in power, you inject the LOX into the fuel flow after it's been heated by the reactor but before it leaves the engine. When you do this, you not only keep all the energy the reactor has imparted to the hydrogen, but the hydrogen and oxygen combust to produce a gain in energy of approximately 200% - hopefully enough to get you off the ground. By very carefully calculating the proportions of fuel and oxidizer you take, you will hopefully be able to launch at high thrust, then return to a more efficient mode to traverse the long distances in space.
Or, to return to our earlier analogy, you can stuff a Nitrous Oxide injector system into the Prius. When you hit the NOS button, the Prius' gas engine will put out double or triple the power - but only for a few seconds, until that NOx tank runs out. That, however, might be enough to let the Prius tow the trailer up the driveway.
There is an even more capable design which pairs a LANTR with a system which utilizes reactor heat to drive a closed-cycle gas turbine, even when the reactor is idling. This would let the system turn a generator and provide electrical power to the ship with no fuel consumption (unlike fuel cells, which 'burn' hydrogen and oxygen). There is a proposed design for an engine of this tri-mode type named TRITON.