A non-causal system is just opposite to that of causal system. If a system depends upon the future values of the input at any instant of the time then the system is said to be non-causal system.
Let us take some examples and try to understand this in a better way.
a) $y(t) = x(t+1)$
We have already discussed this system in causal system too. For any input, it will reduce the system to its future value. For instance, if we put t = 2, it will reduce to x(3), which is a future value. Therefore, the system is Non-Causal.
b) $y(t) = x(t)+x(t+2)$
In this case, x(t) is purely a present value dependent function. We have already discussed that x(t+2) function is future dependent because for t = 3 it will give values for x(5). Therefore, it is Non-causal.
c) $y(t) = x(t-1)+x(t)$
In this system, it depends upon the present and past values of the given input. Whatever values we substitute, it will never show any future dependency. Clearly, it is not a non-causal system; rather it is a Causal system.