Periodic Signal
A continuous time (CT) signal is periodic if it there exists some T such that x(t+T)=x(t) for all t. A discrete time (DT) signal is periodic if there exists some integer N such that x[n+N]=x[n] for all n.
An example of a perdiodic CT signal would be sin(t). T in this instance could be any multiple of $ 2\pi $, so we'll use just plain old $ 2\pi $ as an example.
So, for $ T=2\pi $,
$ \sin{2\pi} = 0 $
$ \sin{4\pi} = 0 $
$ \sin{6\pi} = 0 $
...and so on. You get the idea.
Non-Periodic Signals
Following the above, one example of a non-periodic CT signal would be a logarithmic function $ \log{x} $. In this example, as x increases or decreases the log of x never repeats. This makes the function non-periodic.
