Memoryless System
A system is classified as memoryless if at any given $ t_o\, $, the output $ y(t)\, $ at $ t_o\, $ depends only on the input at the time $ t_o\, $
Example
$ y(t) = 2x(t) + (t-1)^2 \, $
The system is memoryless because it only depends on the input $ x(t)\, $at current time. The $ t-1\, $ is just the value of the time when a signal is inputted into a system
System with Memory
A system is classified as system with memory if the output $ y(t)\, $ at $ t_o\, $ depends on the input at the any time, whether it's in the future or in the past.
Example
$ y(t) = 2x(t-1) \, $
The system is a system with memory because it depends on the input $ x(t-1)\, $at time $ t_o = 1\, $.
