On the Definition of Posets... In my book, the author defines posets formally in the following way:
Let $P$ be a set, and let $\le$ be a relationship on $P$ so that,
$a$. $\le$ is reflective.
$b$. $\le$ is transitive.
$c$. $\le$ is antisymmetric.
Say for $a$, does this merely mean that if some element $x\in P$, $x$ should always have the same relation to itself? and for $b$ if $x$ has the relation to $y$ and $y$ has the relation to $z$, this implies that $x$ has the relation to $z$?

Moreover, when trying to determine if a something is a poset,do I just have to determine if such a relationship exists? And that relationship is not necessarily the usual meaning of "$\le$"


Your statements are correct.
On your last question: A poset is a pair ($P$, $R$), where $P$ is a set and $R$ is a relation on $P$ (which must have properties a, b and c). So the relation is a part of the poset, there is no freedom to choose it.
Hence to determine if something is a poset, you don't have to determine if a suitable order relation exists (it always does), but if the set together with the given relation is a poset. The relation should always be clear from the context, in particular if the relation symbol is not ''$\leq$''.
