I want to show that a one dimensional $\mathfrak {g}\!-\!\operatorname{module}$ is irreducible.
My problem comes down to undestanding the basis of this object. Now assuming the basis works in the same way that the basis of a vectorspace works, I imagine the following argument works:
Let $U$ be a submodule of $V$, where $V$ is our $\mathfrak g\! - \! \operatorname{module}$, and let $V$ have the basis $\{v\}$. Then the submodule $U$ has basis $\{v\}$ or $\{0\}$, hence $U=V$ or $U$ is the zero ideal, and hence $V$ is irreducible.
But perhaps the basis works in a different way. I think my confusion stems from how frequently we alternate between talking about modules and representations.
Yes, it can be confusing talking about modules one moment and then talking about representations the next. In your case there isn't much difference. In fact, the confusion shouldn't confuse you.
A Lie algebra module is a vector space such that ...
A Lie algebra module is irreducible is there are no invariant (non trivial) proper submodules.
Since the dimension is one, there are no (non-trivial) proper submodules. So there isn't really anything to prove.