What is the meaning ‘uniformly converge’？ Assuming that, we randomly sample $n$ data following a distribution, then if someone claims that the average of these $n$ data uniformly converge to its expectation with rate $O(\sqrt{1/n})$. Here, what is the meaning of the 'uniformly converge'?

The expectation of a random variable is just a number, whereas uniform convergence refers to a sequence of functions. The author probably meant almost surely, that is:
$P(\lim_{n\to \infty}\bar X_n=E(X))=1$ which means that $\forall \epsilon>0, \;\exists k\in \mathbb{N}: |\bar X_n-E(X)|<\epsilon \; \forall n>k$ (except for some set of infinitely long sequences of observations whose collective probability of happening is $0$).
