Memory-less System
A system is memory-less if at any given t, the output of the system depends only on the current input at t. The output does not depend on the past or future. You can also think of a system that does not "remember" the past, and cannot "see" into the future.
A system with memory
A system has memory if at any given t, the output of the system depends on the past or future. At a given t, the output relies on outputs of previous t and/or future t.
