As you have seen in the lab, precipitation reactions have a characteristic "look" to them. A solid that was not present before appears in the solution. This solid is one (or sometimes both) of the products of the reaction. When no solid appears, no reaction has taken place.
To understand what is happening on the level of atoms it is important to remember that the reactants are ionic compounds and act as electrolytes in water. That means ions are present. So a solution of silver nitrate doesn't actually contain AgNO3, but rather contains Ag+ and NO3- ions. A conductivity test confirms the presence of these ions.
Thus when silver nitrate solution is added to, let's say, sodium chromate solution (containing Na+ and CrO42- ions), the reaction which occurs takes place among the ions, not the groups of atoms written as formulas on the containers. This is true for all precipitation reactions.