The definition is given by the following formula:
$$\limsup_{n \rightarrow \infty} x_{n} := \lim_{n \rightarrow \infty}(\sup_{m\geq n} x_{m}) $$
I could not understand the meaning of $(\sup_{m\geq n} x_{m})$ and why we have this number m and why it must be $\geq n$, could anyone explain this for me please?
It is just the limit of the sequence $\sup \{x_1,x_2,...\}, \sup \{x_2,x_3,...\},\sup \{x_3,x_4,...\},...$.
Note that the sequence is non increasing (since it is the $\sup$ of a smaller and smaller set), hence it has a limit (or is $\pm \infty$).
For example, let $x_n = (-1)^n (1+{1 \over n})$. Then $\limsup_n x_n = 1$.
Another example: Let $x_n = \begin{cases} 1, & \text{if } n \text{ is a power of }2,\\ -n, & \text{otherwise}\end{cases}$, then $\limsup_n x_n = 1$.