Memory:

Memory is a property relevant only to systems whose input and output signals have the same independent variable. A system is said to be memoryless if its output for each value of the independent variable is dependent only on the  input signal at that value of independent variable. For example the system with description : y(t) = 5x(t) ( y(t) is the output signal corresponding to input signal x(t) ) is memoryless. In the physical world a resistor can be considered to be a memoryless system (with voltage considered to be the input signal, current the output signal).

By definition, a system that does not have this property is said to have memory.

How can we identify if a system has memory?

For a memoryless system, changing the input at any instant can change the output only at that instant. If, in some case, a change in input signal at some instant changes the output at some other instant, we can be sure that the system has memory.

Note: Consider a system whose output Y(t) depends on input X(t) as:  Y(t)   =    X(t-5)     +    { X(t) - X(t-5) }

While at first glance, the system might appear to have memory, it does not. This brings us to the idea that given the description of a system, it need not at all be the most economical one. The same system may have more than one description.