When silver nitrate (AgNO$_3$) reacts with ammonium chloride (NH$_4$Cl), a chemical reaction takes place, resulting in the formation of silver chloride (AgCl) and ammonium nitrate (NH$_4$NO$_3$). The reaction can be represented by the following chemical equation:

AgNO$_3$ (aq) + NH$_4$Cl (aq) → AgCl (s) + NH$_4$NO$_3$ (aq)

In this reaction, silver ions (Ag+) from silver nitrate combine with chloride ions (Cl-) from ammonium chloride to form silver chloride, which is a white, insoluble precipitate. The ammonium ions (NH$_4^+$) and nitrate ions (NO$_3^-$) from the two reactants combine to form ammonium nitrate, which remains dissolved in the solution.

The reaction between silver nitrate and ammonium chloride is a classic example of a precipitation reaction, where two soluble reactants combine to form an insoluble product. This type of reaction is commonly used in qualitative analysis to identify and separate different ions in a solution.