Memoryless System
A memoryless system is, as the name implies, a system that depends only on the present input. For example, the output of the system at any time t depends only on the input at that time t. Any output or input from the time t -1 has no effect whatsoever on the output at time t.
System with Memory
A system with memory is the opposite of a memoryless system. Therefore, the output of the system at any time t is dependent on some past result. For example, if the output of a system, y(t) = x(t) + y(t-1), it is a system with memory.
