Technically, NEMO is a precursor to KM3NeT, which will search for neutrinos from distant sources such as gamma ray bursts or supernovae, as well as the elusive dark matter.
Why do physicists think it's a good idea to build a neutrino experiment underwater in the first place? For the same reason the folks at IceCube wanted to build a large array of detectors deep in the ice in Antarctica.
Neutrinos are extremely difficult to detect, because they very rarely interact with any type of matter, even though they're the most abundant type of particle in the known universe. Only one out of every 1,000 billion solar neutrinos would collide with an atom on its journey through the Earth. And it's easy to confuse them with cosmic rays.
The polar ice is very pure, optically transparent, and free of radioactivity, which can add a lot of extra background "noise" to the data. These qualities make the ice of Antarctica the ideal medium for neutrino detection. Similar properties can be found in the waters of the Mediterranean.
In fact, KM3NeT is intended to complement the ongoing IceCube experiment. IceCube is in the Southern Hemisphere and hence can only detect sources in the Northern sky. NEMO, and later KM3NeT, will be able to cover the rest of the sky.
Both IceCube and KM3NeT are designed to detect Cerenkov radiation as an indicator of a neutrino colliding with an atom in the ice, or sea water - the faint blue glow commonly emitted from radioactive materials when they're immersed in water (as shown in the blue glow surrounding the Advanced Test Reactor, top).
The underlying principle behind Cerenkov radiation is similar to that of a sonic boom. If an aircraft is traveling faster than the speed of sound, the air flowing around its wings can't move out of the way fast enough, creating a sudden pressure drop moving away from the wing at the speed of sound. And this pressure front, or shock wave, creates that loud boom you hear after an aircraft flies overhead.
With Cerenkov radiation, the same kind of shock wave is created, only with light instead of sound. Think of it as a "photonic boom." It occurs when a charged particle moves through a medium faster than the speed of light and generates a similar shock wave.