I will put part of the sentence here because I am interested in something very specific. It follows that
Let $I\subset \mathbb{R}$ be an interval. Define the random variables
$$ E_n(I)=\frac{\#\left( \{\lambda_1(\mathbf{X}_n),...,\lambda_n(\mathbf{X}_n)\}\cap I\right)}{n}. $$
What $\#$ operator does mean? Does $E_n(I)$ represents a sequence of the eigenvalues average? What is the meaning of the $\{\lambda_1(\mathbf{X}_n),...,\lambda_n(\mathbf{X}_n)\}\cap I$?
Since the title of your question references the semicircle law, I assume that $X$ is a random Hermitian matrix. $\{\lambda_{1}(X),\ldots,\lambda_{n}(X)\}$ is the spectrum of the matrix $X$, its set of eigenvalues in $\mathbb{R}$. Then $\{\lambda_{1}(X),\ldots,\lambda_{n}(X)\}\cap I$ is the set of the eigenvalues of $X$ that also lie in the interval $I$. The $\#$ operator takes the cardinality of the subsequent set, so $\#\{\lambda_{1}(X),\ldots,\lambda_{n}(X)\}\cap I$ is the number of eigenvalues of $X$ (without multiplicity) that lie in the interval $I$.
