Marine snow is basically the fall and accumulation of organic debris - fecal pellets, dead plankton and algae, etc. - that are gradually falling to the ocean floor. Since it is overwhelmingly composed of carbonate, in most areas of the ocean pressure, temperature, and chemical composition of ocean water (which has to do with the age of the water mass) cause this fall of marine snow to be dissolved before it gets to the bottom of the sea. (This is is called the carbonate compensation depth, or CCD.) In other areas, however, accumulation occurs, as can see on this pillow basalt. Such accumulation has given us the great chalks of the world, particularly the Cliffs of Dover, as well as many of our oil reservoirs. It's likely the plastic in the phone or computer you're reading this on originated from a large accumulation of marine snow in one of the great shallow basins of the world.
Marine snow represents the largest carbon sink on the planet as well as a a huge food source for organisms that don't live in the uppermost reaches of the water column. One of the most important (and terrifying) effects of a warning trend in climate change is to change the depth at which the CCD occurs - affecting ecosystems around the planet for food availability and ocean acidification, and also affect ocean circulation.
Humorously, marine snow is also associated with "sea snot".