For example
Is $n$ polynomially larger than $\frac{n}{\log n}$? Than $n \log n$?
Is $n^2$ polynomially larger than $\frac{n}{\log n}$? Than $n \log n$?
I am trying to understand the difference because apparently the first line isn't, but the second is (Master Theorem).
"Polynomially larger" means that the ratio of the functions falls between two polynomials, asymptotically. Specifically, $f(n)$ is polynomially greater than $g(n)$ if and only if there exist generalized polynomials (fractional exponents are allowed) $p(n),q(n)$ such that the following inequality holds asymptotically: $$p(n)\leq \frac{f(n)}{g(n)}\leq q(n)$$
For the first problem, we have the ratio is equal to $\log(n)$. It is not the case that there exist polynomials $p(n),q(n)$ such that $p(n)\leq \log(n)\leq q(n)$ asymptotically, because no polynomial is a lower bound for $\log(n)$. Thus it is not polynomially bounded. $n\log(n)$ is the same (even the same quotient if taken in the other order).
For the second problem, we have the ratio is equal to $n\log(n)$. It is the case that $n\leq n\log(n)\leq n^2$ asymptotically, so it is polynomially bounded and therefore $n^2$ is polynomially larger. $\frac{n^2}{n\log(n)}=\frac{n}{\log(n)}$, and we have that (asymptotically) $$n^\frac{1}{3}\leq \frac{n}{\log(n)}\leq n$$