What does "maximum value" of a set of random variables mean? In our statistical mechanics lecture the professor said something along the lines of:

If we have some independent random variables $x_1,x_2,x_3,...,x_n$ having identical distributions:
Suppose $M_{n}=\text{max}(x_1,x_2,x_3,...)$, then we say that
  probability that $M_{n}<x$ is $\text{$\text{Pr}(M_{n}<x$)}$ (say).
In such a case
  $\text{$\text{Pr}(M_n<x$)}=\text{$\text{Pr}(x_1<x,x_2<x,x_3<x,...$)}=(\text{Pr}(x))^{n}$

Now, first of all I don't understand what he meant by $M_{n}=\text{max}(x_1,x_2,x_3,...)$. What does maximum of a set of random variables even mean? Does it refer to the random variable which can take the highest value?
Secondly I don't understand the step $\text{$\text{Pr}(M_n<x$)}=\text{$\text{Pr}(x_1<x,x_2<x,x_3<x,...$)}=(\text{Pr}(x))^{n}$

I think it will help you to think through an example. Consider this one:
Let $x_1, x_2,$ and $x_3$ be three rolls with a fair die. Each $x_i$ can take the values from $1$ to $6$, however it is random which values they take.
Three Examples:
You roll $1$, $5$, and $2$ (i.e. $x_1 = 1$, $x_2 = 5$, $x_3 = 2$). Then $\max\{x_1,x_2,x_3\} = 5$.
You roll $1$, $1$, and $2$. Then $\max\{x_1,x_2,x_3\} = 2$.
You roll $3$, $2$, and $6$. Then $\max\{x_1,x_2,x_3\} = 6$.
Let us now calculate $\mathrm{Pr}(M_3<x)$ for, say, $x=4$. This means that all rolls must be below 4, i.e. 3 or lower. Each roll has $3/6 = 1/2$ chance of rolling $3$ or lower and the rolls are independent, hence the probability is given by
$$\mathrm{Pr}(M_3<4) = \frac12\times\frac12\times\frac12 = \frac18.$$
In other words
$$\mathrm{Pr}(M_3<4) = \left(\mathrm{Pr}(x_i<4)\right)^3. $$
If this example helps you understand it, it may help you generalize to other random variables.
