I know that the Nash equilibrium is state from which if a player will think of moving to another choice he will be not benefited. But what the following notation mean? (Note—I am finding Game Theory notations little difficult so i want explanation with some example.)
DEFINITION 14.1. A Nash equilibrium of a strategic game $\langle N, (A_i), (\succsim_i) \rangle$ is a profile $a^* \in A$ of actions with the property that for every player $i \in N$ we have $$ (a^*_{-i}, a^*_i) \succsim_i (a^*_{-i}, a_i) \qquad \text{for all $a_i \in A_i$}. $$
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Let's unwrap the notation. (In what I follow, I use "strategy" for "profile" when referring to the tuple of strategies of the $n$ players. I have seen used both interchangeably.)
For $1\leq i\leq n$, $A_i$ is the set of possible actions/strategies for player $i$. Thus, a strategy $a$ belongs to the set $A_1\times A_2\times\dots\times A_n$.
For two strategies $a,b$ of $n$ players, and $1\leq i\leq n$ $$ a \gtrsim_i b $$ means that the payoff for player $i$ is greater or equal under strategy $a$ than under strategy $b$.
A game is given by a triple $(N, (A_i)_{1\leq i\leq n}, (\gtrsim_i)_{1\leq i\leq n})$ which specifies the players, the possible strategies for each of them, and the preference relations between those strategies for each player.
By convenient, given a strategy $a = (a_1,\dots, a_{i-1}, a_i, a_{i+1}, a_n)$ and an index $i$, we write $a_{-i}$ for the "substrategy" $$ a_{-i} = (a_1,\dots, a_{i-1}, a_{i+1}, a_n) $$ i.e., "the same thing without the strategy $a_i$ of the $i$-th player"; and given a replacing strategy $\alpha\in A_i$ for this player, $$( a_{-i}, \alpha) = (a_1,\dots, a_{i-1},\alpha, a_{i+1}, a_n)$$ denotes the strategy obtained by replacing, in $a$, the strategy $a_i$ of the $i$-th player by an alternative strategy $\alpha$ (and keeping all the other players' strategies unchanged).
Thus, writing (for $i,a,a^\ast$ as defined in the statement) $$ (a^\ast_{-i}, a^\ast_i) \gtrsim_i (a^\ast_{-i}, a_i) \tag{1} $$ is exactly the same as writing $$ a^\ast \gtrsim_i (a^\ast_{-i}, a_i) \tag{2} $$ and means that, if every player conforms to strategy $a^\ast$, the $i$-th player gains nothing by changing unilaterally their strategy from $a^\ast_i$ to $a_i$ (again, while all $j\neq i$ keep their strategy $a^\ast_j$), as doing so will not increase player $i$-th payoff.