Memoryless Systems
A memoryless system is one that depends only on the current input and is not affected by the past or future inputs. A good example would be the function $ y(t) = [x(t)]^2 $. Here the output $ y(t) $ depends only on the input $ x(t) $ at time $ t $.
Systems With Memory
A system with memory is one which does not fit the definition above for a memoryless system. In other words it has a dependence on past or future inputs. For example, the system $ y(t) = 2x(t) - x(t-1) $ has memory because the output $ y(t) $ depends on the input $ x(t) $ not only at time $ t $ but also at time $ t-1 $.
