Difference between $\mathbb{Q}(x)$ and $\mathbb{Q}[x]$? In general I'm having a hard time understanding what exactly $\mathbb{Q}(x)$ is.... i.e. () circular brackets.

$\mathbb{Q}[x]$ denotes the intersection of all $\textbf {rings}$ containing $\mathbb{Q}$ and $x$, while $\mathbb{Q}(x)$ denotes the intersection of all $\textbf{fields}$ containing $\mathbb{Q}$ and $x$. 
If $x$ is algebraic over $\mathbb{Q}$, you can prove that $\mathbb{Q}[x]=\mathbb{Q}(x)$.
