Terminology: Upper limit and lower limit. Let $(x_n)$ be a sequence of real numbers. It defines:
$\limsup\limits_{n \rightarrow \infty} x_n=\lim\limits_{n \rightarrow \infty} (\sup_{m≥n} x_m)=\inf_{n≥1}(\sup_{m≥n}x_m)$
How should I understand the notation of the definition? Specifically the $m≥n$ and $n≥1$ terms.

The notation $\sup_{m\geq n} x_m$ is just a short way to write $\sup\{x_m: m\geq n\}=\sup\{x_n,x_{n+1},x_{n+2},...\}$, the supremum of the sequence $(x_m)$ starting from the $n$th element.
Similarly, if we let $y_n=\sup_{m\geq n} x_m$ for every $n\in\mathbb{N}$, then $\inf_{n\geq 1} y_n$ is just $\inf\{y_n: n\in\mathbb{N}\}$.
