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In this single-displacement reaction, silver nitrate will dissolve copper, producing elemental silver crystals. The dissolved silver ions in solution attract electrons from the copper, causing the copper to become dissolved copper ions. When the dissolved silver ions gain these electrons, they become undissolved silver atoms, coming out of solution, and crystallizing.
Copper losing its electrons in this process is considered to be oxidizing. Silver gaining electrons in this process is considered to be reducing. Hence, this is a type of reduction-oxidation reaction, or "redox" reaction.
If you look closely, eventually a slight blue color begins to form near the penny. This is the presence of the dissolved copper II ion (Cu+2) which is blue in color.