The primary use of nuclear power for propulsion has always been in submarines, where submerged endurance and detection avoidance are both critical.
Prior to nuclear propulsion, submarines were essentially surface ships that could submerge for brief periods of time. These subs used large storage batteries to travel while submerged. The batteries would discharge, and at that time the submarine would either have to surface or use a snorkel, at which point air could be drawn into the ship so that diesel generators could recharge the batteries.
Submarines' reliance on the internal combustion engine was not convenient from a wartime perspective - it severely limited both submerged endurance and detection. For submarines, nuclear propulsion increased the operating envelope a thousand-fold.
The other chief use of nuclear propulsion is in aircraft carriers. The reason nuclear propulsion is chosen for aircraft carriers is that fuel storage that would otherwise be required for propulsion can now be made available for aviation fuel. Using nuclear propulsion on the largest ship in an aircraft carrier group also reduces the number of refueling ships required, thereby extending the range and/or endurance of the group.
A handful of cruisers were fitted with nuclear propulsion plants, but they are no longer being built. It is much less expensive to build and maintain a ship powered by gas turbines than a nuclear-powered ship, and the unique needs of submarines and aircraft carriers do not apply to other ships - thus these other ships are no longer being built with nuclear propulsion plants.
We have discussed the primary coolant/secondary steam system design, which is the typical propulsion plant arrangement in a nuclear-powered maritime vessel.
What a lot of people aren't aware of is just how eager scientists and engineers were to utilize nuclear energy for other types of propulsion, back in the early days of nuclear power. These engineers envisioned, developed, and tested a number of different technologies for harnessing the atom for propulsion. Let's take a look!
Below is a fascinating five-segment video from the Discovery Channel about the US Air Force's attempt to build a nuclear-powered bomber.
The US tested a number nuclear-powered aircraft engines. Below are two of them (courtesy of Wikipedia):
Experimental HTRE reactors for nuclear aircraft, (HTRE 3 left and HTRE 1 right) on display at Idaho National Laboratory near Arco, Idaho
43°30′42.22″N 113°0′18″W / 43.5117278°N 113.00500°W43.5117278; -113.00500)
These clunky machines are just test beds, and would certainly need to be scaled down for aircraft use!
Nuclear Propulsion in space:
The Air Force wasn't the only entity that wanted to move things around with nuclear power. NASA wanted to make a nuclear rocket motor. Here is a cutaway of a NERVA (Nuclear Engine for Rocket Vehicle Application) motor.
Below, a test platform for a NERVA type rocket motor. Unlike the nuclear aircraft engines, these were very powerful and successful designs that could have been flown, although probably not on manned missions. The one below has not been fired yet, or these people would not be able to approach this closely.
Even docking with one of these briefly (Space Station Re-supply, for example) would be hazardous from a radiation standpoint.
According to wikipedia:
Building on the KIWI series, the Phoebus series were 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.
And lastly there is this... propulsion based on a series of nuclear bomb blasts, at which point we have entered the realm of science fiction! Exept that these have actually been explored as possibilities :)