A displacement reaction occurs when a more reactive substance takes a less reactive one out of its salt solution; a double displacement reaction occurs when two compounds exchange ions.
In a displacement reaction, only a single displacement takes place, whereas, in a double displacement reaction, two displacement takes place between the molecules.
e.g.
Single displacement reaction
Mg + 2HCl → MgCl2 + H2
Double displacement reaction
2KBr + BaI2 → 2KI + BaBr2