http://mathonline.wikidot.com/the-number-of-distinct-relations-on-a-finite-set
From the link above I'm having trouble understanding how from $n^2$ ordered pairs which are either true or false there are a total of $ {2^{n^2}} $, would it not be $2n$?
Also, can someone explain the subset part of the definition, say I have a set $\{a,b,c\}$, would that mean $\{(a,b), (b,a), (c,c)\}$ is also a relation? Or is are the subsets only the pairs of elements of $\{a,b,c\}$?
No, it is correct. Every relation is a subset of $X\times X$ which has power $n^2$. But the number of subsets in a set with $k$ elements is $2^k$, so in our case $2^{n^2}$.
Yes, the set $\{(a,b), (b,a), (c,c)\}$ is a subset of $X\times X$ and thus a relation (which is symmetric).