a) Those reaction in which one element takes the place of another element in compound, are known as displacement reactions.
Example: $$CuSO_4(aq) +Zn(s) \rightarrow ZnSO_4 +Cu$$
When a strip of zinc metal is placed in copper sulphate solution, then zinc sulphate solution and copper are otained.In this reaction ,zinc displaces copper from copper sulphate compound so that copper is set free.The blue colour of copper sulphate solution fades due to the formation of zinc sulphate.
b) Those reactions in which two compounds react by an exchange of ions to form two new compounds are called double displacement reactions.
Example: $$AgNO_3 (aq) +NaCl (aq) \to AgCl + NaNO_3$$
When silver nitrate solution is added to sodium chloride solution, then a white precipitate of silver chloride is formed along with sodium nitrate solution.In this reaction , two compounds silver nitrate and sodium chloride, react to form two new compounds, silver chloride and sodium nitrate.