Memoryless System
A system is said to be memoryless if its output for each value of the independent variable at a given time is dependent only on the input at that same time.
Example:
$ y[n] = (2x[n] - x^2[n])^2 $
As the value of $ y[n] $ at any particular time $ n_0 $ depends only on the value of $ x[n] $ at that time.
System w/ Memory
A system with memory corresponds to the presense of a mechanism in the system that retains information about input values at times other than the current time.
Example:
$ y(t) = \frac{1}{C} \int_{t}^{\infty} x(\tau) d\tau $
