What does apostrophe as a suffix denote? I was just curious as to what "$'$" denotes; i.e. $x' = y$, as in $x'(t) = x(t)$ which has the solution $x(t) = c_1\;e^t$.

I've found out that it has something to do with differential equations, but I can't seem to find any information specifically on "$x'$".

If someone could provide a source to such information, it would be much appreciated.
Typically, $x'$ refers to some derivative of $x$ with respect to a given variable. It is often used in contexts where the derivative being taken is clear, for ease of notation. 

In the equation you list, for example, $x'(t) = x(t)$ is the same thing as writing $\frac{d}{dt}x(t) = x(t)$.
It is a symbol of derivative, it means $\displaystyle \frac{dx}{dt} = y$. You can find the solution by separation of variables technique.