This report has (1) a volatile reference and (2) an insufficiently precise reference. Author must fix. - RBL
Although used mainly in the military as submarines and aircraft carriers, nuclear powered ships have a wide range of applications, and many possible areas of use. Many times, they are more desirable than ships with other sources of fuel, for nuclear powered ships provide more miles per unit of raw fuel compared to combustion-driven power sources, they are much faster, need to carry much less fuel, and do not need an oxygen source. They are also environmentally friendly, for they do not produce greenhouse gasses, and the amount of radiation can be reduced by better containment. Currently there are about 200 nuclear powered vessels operational, and around 700 have been used since the 1950s. 
Nuclear powered ships have been around for quite a while. Research on the subject started in the 1940s, and the first nuclear powered submarine, USS Nautilus, was completed in 1955. This marked a remarkable change in the capabilities of underwater vessels: nuclear powered submarines could sustain 20-25 knots for weeks without surfacing. USS Enterprise, an aircraft carrier followed Nautilus, and in less than a decade, in 1962, the US Navy had 26 functioning submarines, and 30 more under construction. 
The United States and Russia were the first and leading countries to use this technology, with its peak at the end of Cold war: there were around 400 nuclear powered submarines operational or under construction. Due to weapons reduction programs, this number has decreased dramatically, but other countries, such as UK, France, China, and India, have also started using this technology. Still, the US and Russia are still leading countries in using this technology. Today, almost all US aircraft carriers are nuclear powered. 
As for civilian use, nuclear propulsion is essential in icebreaking in the Russian Arctic. Power levels of breaking thick ice, and refueling problems are factors that other types of vessels cannot overcome. Thanks to nuclear technology, Arctic navigation has increased from two to ten months per year, and year round in Western Arctic. The first icebreaker was Lenin, with three 90 MWt OK-150 reactors which were later replaced by two 171 MWt OK-900 reactors which could generate 34 MW of electricity at the propellers. 
Nuclear merchant ships, overall, have not been successful. Although there are a few working examples, their main reason of failure was because of economical reasons. Nuclear powered merchant ships tend to not be economically viable, and thus are not preferred.
Naval reactors are generally the pressurized water type, and differ from other reactors in a couple of ways: they run on highly enriched uranium (>20% U-235, originally 97%, but 93% in latest US submarines, 20-25% in some western vessels, 20% in the first and second generation Russian reactors (1957-81), 45% in 3rd generation Russian units, and 40% in India's Arihant), the fuel is a uranium-zirconium or uranium-aluminum alloy, or a metal-ceramic, they have long core lives therefore need very little refueling, they contain a safe yet compact pressure vessel, and soluble boron is not used. Also, due to flexible power output need and space constraints, the thermal efficiency of naval reactors are less than civil nuclear power plants. 
The United States Navy has had an accident-free record up to this day, operating for 6200 reactor-years and over the course of 230 million kilometers, thanks to excellent training program, and high levels of standardization.  On the other hand, Russia has suffered from a couple of accidents: five where the reactor was irreparable, and others that resulted in radiation leaks. In the late 1970, Russia also stepped up its safety standards, and thus prevented future accident risks.
© Melis Tekant. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
 "Nuclear-Powered Ships," World Nuclear Association. - This reference is volatile. - RBL
 C. R. Nichols and R. G. Williams, Encyclopedia of Marine Science (Facts on File, 2009). - This reference is insufficiently precise. Need article title and page number. - RBL