What is the nature of the particles remaining in the watersolution after reacting with Silver Nitrate and Cu?

Answer 1

AgNO3 + Cu --> Ag + CuNO3
The particles are precipitated silver metal (finely divided), which is displaced from silver nitrate in the above reaction.

Answer 2

If you're using tap water, the silver reacts with the chloride in the water and forms silver chloride, a white cloudy precipitate. You could also be forming copper chloride.