Silver nitrate and sodium chloride are both soluble ionic compounds, which means that they actually exist as ions in aqueous solution.
AgNO_(3(aq)) -> Ag_((aq))^(+) + NO_(3(aq))^(-)AgNO3(aq)→Ag+(aq)+NO−3(aq)
and
NaCl_((aq)) -> Na_((aq))^(+) + Cl_((aq))^(-)NaCl(aq)→Na+(aq)+Cl−(aq)
When these two substances are mixed, the silver cations, Ag^(+)Ag+, and the chloride anions, Cl^(-)Cl−, will bond together to form silver chloride, AgClAgCl, an insoluble solid, which precipitates out of solution.
The other two ions, Na^(+)Na+ and NO_3^(-)NO−3, will continue to exist as such in solution, since the potential compound sodium nitrate, NaNO_3NaNO3, is soluble in aqueous solution.
So, the complete ionic equation looks like this
Ag_((aq))^(+) + NO_(3(aq))^(-) + Na_((aq))^(+) + Cl_((aq))^(-) -> AgCl_((s)) darr + Na_((aq))^(+) + NO_(3(aq))^(-)Ag+(aq)+NO−3(aq)+Na+(aq)+Cl−(aq)→AgCl(s)⏐⏐↓+Na+(aq)+NO−3(aq)
The net ionic equation, which you get by eliminating spectator ions, i.e. the ions that are present on both sides of the eqaution, will be
Ag_((aq))^(+) + cancel(NO_(3(aq))^(-)) + cancel(Na_((aq))^(+)) + Cl_((aq))^(-) -> AgCl_((s)) darr + cancel(Na_((aq))^(+)) + cancel(NO_(3(aq))^(-))
or
Ag_((aq))^(+) + Cl_((aq))^(-) -> AgCl_((s)) darr

So, as a conclusion, a double replacement reaction takes place in which the silver cations and chloride anions form an insoluble white solid called silver hcloride, AgCl.