The NERVA had to use liquid $H_2$... But why? The NERVA engine, developed at the late 60ies, was a nuclear thermal rocket developed for vacuum use in space craft. It was supposed to use Liquid Hydrogen, a cryogenic fuel with quite a few issues such as high cost and difficulty in containment that has greater efficiency than typical hydrocarbon fuels.
So here's where the question comes in: Why did NERVA had to use cryogenic fuel to be worth its trouble (so I've heard)? Cryogenic fuels are far bigger trouble in vacuum stages than in launch stages where they have to stay inside the tank without boiling off years on end. What would the specific impulse of a NERVA using traditional RP-1 or another rocket fuel would be? Is regenerative cooling a part of the reason why LH2 was chosen?
 A: As rockets have finite fuel capacity and important parameter is how much rocket velocity can be achieved in a given situation per unit of fuel mass consumed. The higher the exhaust velocity the more effectively the fuel mass is utilised. Issues such as energy required are also important but generally exhaust velocity or "specific impulse" is amongst the most important factors in either duration for which a given thrust can be produced. This translates directly (albeit very non linearly) to eg mass that may be lifted to a given orbit for a given launch mass. 
So, simplistically, the faster mass can be ejected the more impulse can be achieved.
Exhaust velocity (m/s) can also be expressed in terms of "specific impulse" = units of force produced for a given time period for a given mass of fuel consumed.
Specific Impulse = kg thrust / kg mass of fuel consumed. per unit time.
Specific Impulse is often given with units of "seconds" based on cancelling kgf/kgm causing purists to complain severely -  add the required constant "g" and the units are (not unexpectedly) m/s.)
As explained in Wikipedia  (see image below) 
exhaust velocity is inversely proportional to the square root of the the gas molecular mass M in kg.kmol - see arcanish formula below. The lower this figure the better. Hydrogen gas has by far the best figure of merit available. 

So - to make best use of the energy from the nuclear reactor Hydrogen is used as the "reaction mass".  NERVA was effectively a rather large "Hydrogen heater" which takes liquid Hydrogen as input and expels Hydrogen gas. Liquid Hydrogen has a density not much more than gasous air so needs vast tank sizes for a given mass of fuel - and is still the overwhelmingly superior choice in this application.
Kiwi B4A - a genuine working NERVA rocket motor:

Wikipedia - the NERVA family 
Wikipedia - Nuclear thermal rocket
A working NERVA being overdriven to the point of self destruction to test safety and shutdown here
Many NERVA images
A: Nuclear rocket motors work by heating a gas and allowing it to expand out of the exhaust. To get the most thrust from your gas you want the momentum of the gas molecules to be as high as possible, because the force is equal to the rate of change of momentum of the gas molecules.
Suppose the nuclear reactor heats the gas to a temperature $T$, then the kinetic energy of the gas molecules will be approximately related to the temperature by:
$$ \tfrac{1}{2}mv^2 = \tfrac{3}{2}kT $$
and therefore:
$$ v = \sqrt{\frac{3kT}{m}} $$
and the momentum change per molecule is therefore:
$$ \Delta p = mv = \sqrt{3kTm} $$
If we're expelling a mass $M$ of propellant per second the number of molecules being expelled per second is $M/m$, and the total change of momentum is the momentum change calculated above multiplied by the number of molecules being expelled per second:
$$ \Delta p_\text{total} = \sqrt{3kTm} \frac{M}{m} = M\sqrt{\frac{3kT}{m}} $$
So there you have it. The total momentum change per second is just the thrust, and it's inversely proportional to $\sqrt{m}$. To get the maximum thrust you want the lightest gas molecules you can get, and that's hydrogen.
A: Here are some thoughts on using water instead of hydrogen as reaction mass: The complications of storing hydrogen compactly are certainly a factor to weigh against the extra thrust per unit weight achievable with hydrogen. Since water molecule weighs 9 times more than hydrogen molecule (18 vs 2) and the thrust at given temp is proportion to square of mass, water as reaction mass would give three times less thrust all else being equal. Might be worth carrying three times more mass to avoid hassle of handling H2. Also, the volume needed to hold the same weight of fuel and therefore the container size and weight would be less with water. Not to mention the weight of all the equipment for cryogenic cooling and storage. 
